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# Meeting Details

Title: Symplectic Rigidity and Lagrangian Cobordisms Department of Mathematics Colloquium Joshua Sabloff, Haverford College A fruitful way to frame the study of symplectic geometry is as the search for the boundary between symplectic flexibility (when symplectic objects behave like topological objects) and rigidity (when symplectic behavior is more restrictive). I will introduce this central theme through a visually appealing problem involving geometric properties of special cylinders in $R^4$, and will then describe a technique (capacities derived from generating families) for solving this problem. This is joint work with Lisa Traynor.

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Poster
The Sample Complexity of Semi-Supervised Learning with Nonparametric Mixture Models
Chen Dan · Liu Leqi · Bryon Aragam · Pradeep Ravikumar · Eric Xing

Thu Dec 6th 05:00 -- 07:00 PM @ Room 210 #96
We study the sample complexity of semi-supervised learning (SSL) and introduce new assumptions based on the mismatch between a mixture model learned from unlabeled data and the true mixture model induced by the (unknown) class conditional distributions. Under these assumptions, we establish an $\Omega(K\log K)$ labeled sample complexity bound without imposing parametric assumptions, where $K$ is the number of classes. Our results suggest that even in nonparametric settings it is possible to learn a near-optimal classifier using only a few labeled samples. Unlike previous theoretical work which focuses on binary classification, we consider general multiclass classification ($K>2$), which requires solving a difficult permutation learning problem. This permutation defines a classifier whose classification error is controlled by the Wasserstein distance between mixing measures, and we provide finite-sample results characterizing the behaviour of the excess risk of this classifier. Finally, we describe three algorithms for computing these estimators based on a connection to bipartite graph matching, and perform experiments to illustrate the superiority of the MLE over the majority vote estimator.


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# I know C++ 2003 How to start learning C++ 2017?

Tl;dr - the subject is enough. For background, read below.

I am a professional, freelance, embedded software developer. I first learned about C++ in the 90s and taught myself from a book C++ for C programmers.

The beauty of that was that it assumed that I knew C and didn't have to spend chapter after chapter explaining how to declare a variable/what a variable is, function/passing parameters/returning a result, loops, etc, etc, etc.

So, it cut quickly to the chase and I started learning from page 1, rather than skipping forward.

Fast (?) forward a few decades, and I no longer code (much) assembler. Work seems to be split between C, Ada (mostly for defence) and C++.

It has been four years since my last C++ gig, and, industry being conservative does not adopt new version of languages for some time, until they can be seen to be widely accepted as stable (if any of you non-industry guys what to know why, then we generally want to wait until "they get the bugs out of the new compiler". Given that a lot of embedded can be life-critical, you can understand why).

So, now I am seeing ads requiring C++ 2014 and 2017 (and my latest experience is 2003). The delicious irony of this is that I know, from bitter experience, that coding standards will forbid me from using the new features intruded in those versions.

All of my life, I have taught myself new languages & concepts from books. Earlier this year, I took a leap of faith & taught myself Flutter/Dart (for personal, non-professional use) from a Udemy course. That was an eye-opener, and I won't be responsible for the death of any more trees in future.

But, I digress (this entire screed has been a digression, if you ask me). I have now purchased another Udemy course to learn C++.

And, finally, after all of that, my question is still the same as its title - "I know C++ 2003 How to start learning C++ 2017 ?".

Do I watch the start of each chapter, and skip if I yawn? Do I read the table of contents and take a stab at where to jump in?

Given that you are a group of educators, I imagine that this is not a new situation to you, but am not sure that it has an answer which does not involve the length of pieces of string.

• You might browse en.cppreference.com/w/cpp and look at the various C++ versions and features. When you find something interesting, experiment with it and see if you find it useful. – Fritz Sieker Dec 29 '20 at 18:54
• Recommend this lecture by Stroustrup circa 2014 on "The Essence of C++" in which along the way he highlights what he considers canonical C++14 usage: youtube.com/watch?v=86xWVb4XIyE – Daniel R. Collins Dec 30 '20 at 1:00
• I read the list of features that were introduced in each new standard: As I read I smiled an said "aah new feature-a, they took that from Lisp, but a different syntax", and "aah new feature-b, they took that from Lisp, but a different syntax", … In the end my heart sank. I could learn this, but how could I ever hire someone that know it: C++ was already too big, now it is bigger. – ctrl-alt-delor Dec 31 '20 at 23:48
• Well, it has been *cough* decades since I coded Lisp, at uni, but I was thinking "ooh, JS feature envy!" a few times, so I know what you mean (+1). Why bother with promises when you already have callbacks (and Finite State Machines)? Why lambdas, when anything over a 2 or 3 line body is tempting unreadable/unmaintainable code, and anything shorter will be inlined by the compiler? Fortunately, as I mentioned, most company coding standards forbid many of the frills. But, alas, they don't forbid interviewers from asking about them :-( – Mawg says reinstate Monica Jan 2 at 8:41

I had a similar problem 8/9 years ago, but with C++11.

The trick is that there is no trick. You simply sit and use the feature you want to use. Besides that it is also beneficial to know C++14 before getting into C++17 and know C++11 before getting familiar with C++14 features. (Obviously prerequisite for C++11 is C++03/98, but that's already covered by OP).

If getting up-to-date with C++17 (C++20 actually was vote in 2020) I would start with feature list grouped here. The problem is that there are so many of them that one may have problems in how to approach them, thus I've provided an ordered list that shows how I would teach it to people:

## Basic C++11 stuff

• move semantics (overview + SO question) - move semantics and rvalue references are the corner stone of modern C++, what follows is the rule of 5 (or 3/5/0 if being precise)
• auto
• lambda expressions
• ranged for loops - because for(auto const& e: {1, 2, 3}) { std::cout <<e <<'\n'; } is so cool :)
• classes related stuff:
• constexpr specifier
• initializers lists - it is used mainly in std::vector<...> so it's important to know how it works and what are problems involved with them (it is not my favourite C++11 feature)

## C++14 & C++17

• After these basics I would read about the other C++11 related features or moved to C++14 features listed here (and consult C++11 features list when necessary). Note that C++14 was a nice little amendment to C++11 so there aren't many new (and cool) stuff.
• C++17 features worth mentioning (very subjective opinion, at this point you should already know what you want/need to know):

## C++20

• Concepts are the new big thing
• immediate functions
• IMHO modules will be a dead end as long as C++ will be split into header and implementation fiels (besides Rust have better package/modules concept). But that's just an opinion bouncing in my head.
• Remaining C++20 and previous version stuff (obviously only a fraction was covered in this post :)).

I would guess that taking such courses is a gross waste of time unless they are specifically focused on recent additions to C++. Most of the "beginning C++ courses" won't even get you to essential new features.

I haven't followed updates to the language for a long time but first, I think that the "new" language is backward compatible with the "old" language (unlike Python). So, what you need are incremental changes in most things to what you already know.

But I also think that the programming style in modern C++ has changed, depending more on library code than was true in the past. What you need to do is find a way to become comfortable with that newer programming style.

I would think that a book like this one from the very capable Scott Myers would get you started. I don't have the book, but Myers has been trustworthy in the past. Use the ideas there to build something significant.

• I do like Myers, but he has retired form C++. I am not sure why I even asked this, as code, despite being he "be all and end all" is actually the most trivial part of sotware development. Btw, can hands dirty industry guys help you educators? – Mawg says reinstate Monica Dec 29 '20 at 16:49
• @MawgsaysreinstateMonica They help at my institution. They do curricular review with us as part of our long-term planning and oversight. It's very helpful! We also sometimes have professionals come and speak to the students about careers. – Ben I. Dec 29 '20 at 20:45
• Us industry folk can be the problem: Making students learn Java and C++ before they can program. However Lean, test-driven, revision-control, Unix, Free-software, and a few other practices would not go a miss. – ctrl-alt-delor Dec 31 '20 at 23:52


text:
Text.LaTeX.Base.Pretty

Description

LaTeX values pretty printer.

Still experimental. Give it a try and send us your feedback! :)

Synopsis

# LaTeX pretty printer

Pretty print a LaTeX value. It produces a more human-friendly output than render.

This function should be used only for debugging purposes since it may change the semantics of the input in order to create a prettier output. In other words, running a LaTeX compiler in the output file of renderFile fp l may produce a different document than running it in the output of writeFile fp (prettyLaTeX l). You should use renderFile unless you really need to read the LaTeX file.

# Configurable printer

docLaTeX :: LaTeX -> Doc () Source #

This function transforms a value of type LaTeX to a Doc. You can then choose how to print this Doc value using the function from the Text.PrettyPrint.Free module.


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# Field Z/mZ

#### Fernando Revilla

##### Well-known member
MHB Math Helper
I quote a question from Yahoo! Answers

Multiply in Z\mZ question: m= 3, 7 also which m is Z/mZ a field? m=7? m=3? show some steps.?
multiply in Z\mZ
question: m= 3, 7 also which m is Z/mZ a field?
m=7?
m=3?
I have given a link to the topic there so the OP can see my response.

#### Fernando Revilla

##### Well-known member
MHB Math Helper
I suppose you want to prove that if $m$ prime, then $\mathbb{Z}/m\mathbb{Z}$ is a field. For all $m\geq 2$ integer, we know that $\mathbb{Z}/m\mathbb{Z}=\{\bar{0},\bar{1},\ldots,\overline{m-1}\}$ is a finite, conmutative and unitary ring. But we also know that a finite integral domain is a field, so we only need to prove that if $m$ prime, there are no divisors of zero.

Suppose $\bar{k}\bar{s}=\bar{0}$, then $ks$ is multiple of $m$ or equivalently $m|ks$. If $m$ prime, $m|k$ or $m|s$ wich implies $\bar{k}=\bar{0}$ or $\bar{s}=\bar{0}$.

For example, in the particular case $m=7$ the inverses are
$$(\bar{1})^{-1}=\bar{1},\;(\bar{2})^{-1}=\bar{4},\;(\bar{3})^{-1}=\bar{5},\;(\bar{4})^{-1}=\bar{2},\;(\bar{5})^{-1}=\bar{3},\;(\bar{6})^{-1}=\bar{6}$$


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# Why do first and second moments insure stability?

In Pattern Recognition and Machine Learning, Bishop states

"Let us now consider the maximum entropy configuration for a continuous variable. In order for this maximum to be well defined, it will be necessary to constrain the first and second moments of p(x) as well as preserving the normalization constraint"

Why is it the case?

Entropy:

Moments:

Consider the uniform distribution on the interval $$[-x,x], x \ge 1$$ . It's entropy is $$ln(2x)$$ . When x go to infinity , the entropy go to infinity as well, therefore the maximum is not well defined .

• My question is about the moments, why do you consider these moments to prove that it's well defined – ted Feb 23 '19 at 17:38
• Without the moments bounds , as the exemple shows , the entropy may not by finite . Now consider that the random variable has finite variance . The probability distribution with the highest entropy is the uniform distribution ( we restrict our atention to an interval $[a,b]$ ) . The variance of this RV is $1/12(b-a)^{2}$ and it's entropy is $ln(b-a)$ . Therefore a bound on the variance implies a bound on the entropy . – Popescu Claudiu Feb 23 '19 at 19:27
• @PopescuClaudiu But I thought Bishop just showed us that the Gaussian, not the uniform, is the distribution having highest entropy for a fixed variance (and first and zeroth moment)... max entropy for a given variance is given by eqn 1.110. – Don Slowik Apr 16 '19 at 22:42
• @Don Slowik The probability distribution with the highest entropy is the uniform distribution when no restrictions are enforced (I only impose the technical condition that the variance is finite). The Gaussian is the distribution having highest entropy for a fixed variance , as you said . There is no contradiction here . – Popescu Claudiu Apr 17 '19 at 8:38

Many continuous distributions have infinite entropy. So it makes sense to impose some restrictions before asking which distribution has the largest entropy. For example, if we just required the zeroth moment be 1: $$\int^\infty_{-\infty} p(x)dx=1$$, $$\text{H}[\text{Unif}(x|a,b)]=\ln(b-a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(@PopescuClaudiu)}$$ and $$\text{H}[\mathcal{N}(y|\mu, \sigma^2)]={1\over{2}} \{1+\ln(2\pi \sigma^2)\} \ \ \ \ (\text{Bishop eqn. 1.110})$$ can each be made as large as desired by increasing $$b-a$$ or $$\sigma^2$$ respectively.

The continuous distribution having the largest entropy on a fixed domain, $$x\in[a,b]$$, can be found by maximizing $$-\int^b_a p(x)\ln(p(x))dx + \lambda\bigg(\int^b_a p(x)dx-1\bigg)$$ wrt $$p(x)$$: $$-\ln p(x)-1+\lambda=0$$ $$p(x)=\exp(\lambda-1)=c$$ $$\int^b_a p(x)dx=1\Rightarrow p(x)={1\over{b-a}}$$ as pointed out in the comment by @PopescuClaudiu.

The continuous distribution having the largest entropy and having first and second moments of $$\mu={b+a\over2}$$, and $$\sigma^2={(b-a)^2\over{12}}$$ is $$p(x)={1\over(2\pi\sigma^2)^{1/2}}\exp\left\{-{(x-\mu)^2\over2\sigma^2}\right\}$$ as derived in Bishop PRML p.54. Note that each of these distributions have the same $$0^{th}, 1^{st}$$ and $$2^{nd}$$ moments, and the Gaussian has a larger entropy $${1\over{2}} \{1+\ln(2\pi \sigma^2)\}= {1\over{2}} \{1+\ln({\pi\over6})+2\ln(b-a)\}>\ln(b-a)$$

A whole collection of possible constraints and corresponding entropy maximizing distributions is given here.

• Can u explain in detail that how to get from $-\int^b_a p(x)\ln(p(x))dx + \lambda\bigg(\int^b_a p(x)dx-1\bigg)$ wrt p(x) to $-lnp(x) - 1 + \lambda$? – SundayCat Apr 30 '19 at 2:03
• So this is Calculus of Variations which is described in appendix D of Bishop PRML. Basically, the variation of those integrals wrt small, arbitrary changes $\delta p(x)$ should be zero at an extremum. – Don Slowik May 1 '19 at 0:48
• Got it, thanks a lot – SundayCat Jun 14 '19 at 19:18


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# What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a continuum taken more or less as a primitive notion. Modern foundational theories starting around 1870 are based on a continuum made of points and therefore cannot serve as a basis for interpreting the thinking of the earlier mathematicians as far as the foundations are concerned.

What one can however seek to interpret are the techniques and procedures (rather than foundations) of the earlier authors, using techniques and procedures available in modern frameworks. In the case of analysis, the modern frameworks available are those developed by Weierstrass and his followers around 1870 and based on an Archimedean continuum, as well as more recently those developed by Robinson and his followers, and based on a continuum containing infinitesimals, and other frameworks such as the one developed by Lawvere, Kock, and others.

I was therefore a bit puzzled by the following comment by a historian expressed here:

Recently there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern non-standard analysis, in which infinite and infinitesimal quantities do exist. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend in the last few decades. It is parsimonious and requires no expert defence for which modern concepts seem essential and therefore create more problems than they solve (e.g. with infinite series). The same can be said of non-standard readings of Euler; etc.

Question 1. Is this historian choosing one foundational framework over another in interpreting the techniques and procedures of the historical authors?

Question 2. What exactly is a Parsimonious History?

Question 3. Gray and Bottazzini reportedly make a rather poetic proposal in the following terms: "The best policy is to read on in a spirit of dialogue with the earlier authors." The proposal of such a conversation with, say, Euler sounds intriguing. I am only wondering about Gray's comment here that "Euler’s attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak." Isn't such an opening line in a conversation likely to be a conversation-stopper?

Question 4. In connection with the work of Laugwitz mentioned by one of the responders, one could ask why Gray does not cite explicitly the work of any of the authors that he wishes explicitly to criticize for using modern infinitesimals? Laugwitz's article (Laugwitz, Detlef. Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39 (1989), no. 3, 195–245) appeared in Archive for History of Exact Sciences, clearly a reputable journal since Jeremy Gray happens to be its Editor-in-Chief. Similarly, the article McKinzie, Mark; Tuckey, Curtis. Hidden lemmas in Euler's summation of the reciprocals of the squares. Arch. Hist. Exact Sci. 51 (1997), no. 1, 29–57 appeared in the same journal and exploited Robinson's framework to clarify some of Euler's procedures; it, too, is being stonewalled by the Editor-in-Chief.

• It’s impossible to say for sure without seeing the historical interpretation that he offered. It sounds, however, as if he’s arguing against the idea that Leibniz et al. were using an informal version of infinitesimals on the grounds that a defensible, formal version didn’t exist until relatively recently. This is obviously a specious argument. – Brian M. Scott May 30 '16 at 18:10
• @BrianM.Scott I read it more narrowly, that he says arguing that Leibniz et al. were using some informal version of rigorous modern non-standard analysis - such as Robinson's for example - is inadequate. Rather, they had their own concepts of infinitesimals, which, I guess, would have significant differences to any given modern theory (and of course also significant similarities). And one should aim to understand their own concepts when writing their history. Which sounds quite sensible to me. – Daniel Fischer May 31 '16 at 19:11
• @Daniel: I question whether there's a real difference between an informal version of rigorous modern non-standard analysis and any notion of infinitesimals that they may plausibly have had. I don't see that rigorous has any real effect: surely no one would suggest that their notion(s) were close enough to ours for the details implicit in rigorous to be meaningful or relevant. I certainly don't see that there is any conflict between trying to understand their concepts and suggesting that those concepts were to some considerable extent informal versions of concepts that we have formalized. – Brian M. Scott May 31 '16 at 20:38
• That their "concepts were to some considerable extent informal versions of concepts that we have formalized" is more than plausible. But one should try to understand their concepts (as far as one can reconstruct them) independently of the modern theories, and then see whether they correspond to the modern theory's concepts or deviate from them, rather than shoehorning their concepts into the modern theory. – Daniel Fischer May 31 '16 at 21:10
• Can you please stop adding new questions here? After there are already answers to the original question, it's unfair to the answerers to add further questions. Create a new question for the further questions that arose. – Daniel Fischer Jun 1 '16 at 14:29

Comment

What about Detlef Laugwitz's comments in his: Infinitely small quantities in Cauchy's textbooks, Hist.Math. 14(1987), no.3, 258–274 [that you cited elsewhere in SE]:

As a historian of mathematics one cannot but take an author’s own intentions and reasons seriously: Infinitely small quantities are fundamental in Cauchy’s analysis, they are compatible with rigor, and they produce simplicity.

[Cauchy's theorems on continuity and convergence] are incorrect when interpreted in the by now common conceptual framework of analysis (which obviously cannot have been Cauchy’s framework). Both theorems become correct as soon as one adds assumptions on uniformity (which, at least in the form by now common, were never used by Cauchy). The theorems are correct in any of the modern theories of infinitesimals (which, apart from being unknown to Cauchy, lack the “simplicity of infinitesimals,” at least in the version of Robinson).

The three attitudes mentioned (Cauchy erred; Cauchy forgot about essential assumptions; Cauchy was correct, but only when put against a modern background) are unsatisfactory from the point of view of a historian. [...] The only satisfactory attitude should be: Try and understand Cauchy’s theorems and their proofs from his own concepts.

Note: according to my understanding of J.Gray's point of view, a "parsimonious history" is an approach aimed at understanding past theories and concepts "in their original environment", avoiding to overload them with recent developments.

• Thanks for this. Laugwitz clearly disagrees with Gray's thesis here. Would you include Laugwitz's paper in Archive for History of Exact Sciences here as well? – Mikhail Katz May 31 '16 at 16:03
• @MikhailKatz "Try and understand Cauchy’s theorems and their proofs from his own concepts." is exactly how I understand the quote of Gray in your question. – Daniel Fischer May 31 '16 at 19:14
• @DanielFischer, the question is, why should one modern theory as opposed to another be excluded apriori from providing proxies for historical procedures. – Mikhail Katz Jun 1 '16 at 8:01
• Mauro, what I am trying to understand is the precise meaning of "recent developments". Does this refer to foundational issues of justification of the kind of entities the historical authors used (this might be called an ontological issue), or rather the procedures the historical authors used and their proxies in modern theories? – Mikhail Katz Jun 1 '16 at 8:23
• There seem to be unbalanced parentheses in your answer. – Mikhail Katz Jun 3 '16 at 7:33

Note: [2016-06-01] Please note that OP has changed the title which was:

How can a modern historian interpret historical mathematicians?

and he has added question 2. This answer addresses the original question which is at the time question 1.

To me it seems Jeremy Gray, the author of the referenced book, put some words of warning that interpreting the work of the old masters in terms of one or two modern concepts only is too often superficial and not appropriate to fully grasp what was really going on.

If we look at the last sentence of OPs cited paragraph we can see that the historical development is more than that.

From section 1.5: But it is to say that the foundations of the calculus were for at least two centuries the subject of shifting, partial, and largely coherent speculations that form the opening chapters of the history of analysis.

To better see the author's position I like to cite from Hidden Harmony-Geometric Fantasies: The Rise of Complex Function Theory written by Jeremy Gray and Umberto Bottazzini.

From the Introductory Section: In any history of ideas, the historian seeks to show how things once thought about in one way became thought about in another. As complex function theory developed many ideas were first introduced naively and only slowly refined.

Definitions were lacking, and when provided were sometimes inadquate by later standards. Moreover precision, when it became available, could be misleading: mathematicians on occasion offer a clear definition with very few ideas about its deepest implications - as the example of continuity in real analysis shows.

Sometimes these problems can be confronted directly, as with the very definition of an analytic function, but more often one has to ride out a long period of some vagueness.

Let us note some specific issues: Cauchy, for example, often used the phrases continuous and finite and continuous very loosely to mean something like complex analytic. Similar problems occcur with counting roots according to their multiplicities, with $\lim$ versus $\limsup$, and points of infinity and poles. ...

Later on the authors continue with their preferred approach to the history of mathematics and mathematicians:

There is therefore no truly satisfactory way to represent the originial ideas of mathematicians when they are like this. To say nothing is to produce confusion. To silently bring them into line with modern standards not only introduces anachronisms but also brings in historical falsehoods and nullifies the purpose of history.

To correct them in more than the most egregious cases is to encumbeer genuine blunders and thereby diminish the work of major mathematicians.

The best policy is to read on in a spirit of dialogue with the earlier authors, aware, as one might be, of the limitations and false implications of their papers and books, and waiting to see when, if at all in the period, better light was shone on the subject. In this way one can grapple with more of the complexity, and the drama, of the past.

• You (or perhaps the authors of the book from where you quote) have captured the spirit of mathematical development over time. +1 for the nice answer. – Paramanand Singh May 31 '16 at 9:54
• @ParamanandSingh: Perhaps me, but for sure the authors from where I quote. :-) Many thanks for your nice comment. – Markus Scheuer May 31 '16 at 10:09
• What do you think they mean when they claim that "Definitions were lacking"? They go on to say that sometimes definitions were provided, but what does the original comment mean? – Mikhail Katz May 31 '16 at 11:24
• Well the issue of foundations is kind of precisely the problem. What they seem to be saying is that set-theoretic definitions were lacking, though they don't mention this explicitly. I don't see them introducing the distinction between definitions and foundations clearly enough. The example they give, namely that of continuity, is particularly ill-chosen because Cauchy's definition of continuity was actually quite precise. Do they give any evidence in the book that he used "continuous" in the sense of "complex analytic"? I find this surprising. – Mikhail Katz May 31 '16 at 12:09
• Since you are the one who brought it up in your answer can you give a sample of their insights? – Mikhail Katz May 31 '16 at 15:44

This is in response to a claim by Gray and Bottazzini that allegedly "Definitions were lacking". So what? It is a basic (meta)mathematical problem:

• you are given a proof that contains no definitions (or, perhaps, refers to excessively narrow definitions)
• recover the definitions.

This is what Bourbaki was doing when he was recovering real definitions from the boring and repetitive proofs of the early functional analysis.

Why this approach cannot be used as a tool for analysis of the history of mathematics? Mathematicians of the past were using unstated definitions and unstated assumptions. Why are research mathematicians of nowadays are forbidden from doing some kind of reverse engineering and recovering their ancient colleagues' definitions and assumptions?

www.borovik.net/selecta

• "Why are research mathematicians of nowadays are forbidden from doing some kind of reverse engineering and recovering their ancient colleagues' definitions and assumptions?" 1. Who forbids whom of doing that, exactly? 2. What you call reverse engineering is actually often what mathematicians turned historians practice and support, at the expense of what historians view as being history. 3. Are you suggesting that Gray is not a mathematician? – Did Jun 1 '16 at 19:14
• This does not appear to be an answer to the question, which is asking for an explanation of Gray's views, not a rebuttal of them. – Eric Wofsey Jun 1 '16 at 19:54
• @EricWofsey, it can be debated whether this answer is conclusive, but it seems to me that the intention is to provide an analysis of misconceptions behind Gray's views which would explain Gray's comment. – Mikhail Katz Jun 2 '16 at 7:07

Gray's "parsimonious" argument could be termed the Gray sword analogously with the Occam razor, and if applied in the context of a proper focus on procedures would in fact yield the opposite result of the one Gray seeks.

Consider for example Cauchy's definition of continuity, namely an infinitesimal change $\alpha$ in the variable $x$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function. In a modern infinitesimal framework one copies this over almost verbatim to get a precise definition of continuity.

If one wishes to work in a traditional Weierstrassian framework, one needs to interpret Cauchy's definition as "really" saying that, for example, for every epsilon there is a delta such that for every $x$, etc.

Such logical complexity involving alternating quantifiers will surely fall by the (Gray) sword. Alternatively, one could seek to interpret Cauchy by means of sequences, which is not much better because Cauchy explicitly says in defining an infinitesimal that a sequence becomes an infinitesimal (rather than an infinitesimal being a sequence). So apparently Gray should be saying the following:

Since Boyer (at least) there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern Weierstrassian analysis. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend since Weierstrass came along. It is parsimonious and requires no expert defence for which modern alternating quantifiers seem essential and therefore create more problems than they solve.


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# Coleman-Mandula theorem

Post-publication activity

Curator: Jeffrey E. Mandula

The Coleman-Mandula theorem states what Lie group symmetries are possible in a relativistic theory of interacting particles.

## Statement of the Coleman-Mandula theorem

The Coleman-Mandula Theorem concerns what symmetries are possible in a relativistic theory of interacting particles (Coleman and Mandula, 1967). It states that the only possible such Lie group symmetries are direct products of the Poincaré group and an internal symmetry group. If we denote the state of a single particle by $\left| i\lambda p \right\rangle$, where $i$ denotes the particle type, $\lambda$ its intrinsic spin, and $p$ its momentum, an internal symmetry transformation is represented by a matrix acting only on the particle type index while a Poincaré transformation leaves the particle type unchanged and only affects its spin and momentum. The sorts of transformations excluded by the Coleman-Mandula theorem are those whose action on the $i\lambda p$ labels cannot be factored into the product of two terms, one acting only on the particle type index and the other acting only on the space-time indices. We would call such transformations hybrid symmetries.

Two important properties of a symmetry group of a relativistic particle theory are that single particle states form representations of the group, and that multiparticle states transform like direct products of single-particle representations. We will assume that the elements of a symmetry group are unitary operators acting on the space of states.

Corresponding to the unitarity of the group elements, the generators of the symmetry group are assumed Hermitean. The eigenvalues of the generators are the quantum numbers carried by the particles of the theory. The statement that multiparticle states transform like direct products of single particle representations is equivalent to the additive conservation of these quantum numbers. All of the generators of an internal symmetry group commute with each of the generators of the Poincaré group.

The theorem is phrased purely in terms of observable properties of particles and their interactions. Neither the statement of the theorem nor its proof use the full apparatus of quantum field theory. This pedagogical orientation is an artifact of the state of particle theory when the questions that led to its discovery arose. This will be reviewed in the next section of this article.

## Context of the theorem

In the 1950's, experiments at particle accelerators had discovered many new particles and scattering resonances. The masses and spins of these particles were measured, and the patterns of allowed and forbidden decays were observed. These discoveries were new information about the strong, nuclear force, at much higher energies than in previous experiments. They invigorated the attempt to build a theory of the strong interactions.

At that time the only successful theory of elementary particle interactions known was quantum electrodynamics, which seemed to describe all electromagnetic phenomena. This was the natural model from which to start, but it was immediately realized that trying to describe the strong interactions as a quantum field theory was not straightforward. Among the host of obstacles was the fact that there were no reliable methods of calculation in strongly coupled field theory, and that there was no method for choosing a set of fundamental fields.

Without a clear path forward, progress came from attempts to exploit general properties of relativistic quantum field theories. The results that came from such work could be true properties of the strong interactions. One approach was the use of dispersion relations, which express the principle of causality in terms of the analytic properties of scattering amplitudes as functions of complex momenta. Dispersion relations, supplemented by quite reasonable simplifying assumptions, gave good descriptions of many aspects of high energy scattering, such as the electromagnetic form factors of nucleons. Attempts to build a complete theory of the strong interactions were not immediately successful, but the ideas developed in this context have had a powerful and continuing effect on elementary particle theory.

A particularly successful approach was to look for symmetry principles to organize the data. Of course, symmetries alone could not give a complete description of strong interactions, but it was plausible that one could discover the symmetries of the theory underlying the resonances and stable particles before having found the true theory. Furthermore, discovering the symmetries of the theory could be a major step in finding the theory itself.

Symmetry groups had been successfully applied in nuclear physics. The paradigm of an internal symmetry was isotopic spin, which related the nuclei of isotopes of different elements, but with the same atomic number. Isotopic spin was seen to be a symmetry of the strong interactions as well. Resonances with different charges but (almost) identical mass could be grouped into multiplets that formed representations of the isospin group, SU(2), and their allowed decays followed the patterns expected from group representation theory.

Besides isospin, a new internal quantum number was discovered. It was observed that many possible decay modes that were allowed by conservation of energy, charge, and isotopic spin, did not occur at anything like the expected natural strong interaction rate. This puzzle was solved by assigning to each particle a new quantum number, called "strangeness", and positing that it had to be conserved in strong interaction decays (Nakano and Nishijima, 1953; Gell-Mann, 1956). Particles in the same isotopic spin multiplet had the same value of this new quantum number, that is, strangeness commuted with isotopic spin.

The 1960's saw a great expansion of the use of symmetry groups in particle physics. A spectacular development was the observation that resonances with similar masses but different charge, isospin, and strangeness could be collected into multiplets that formed representations of a larger group, SU(3) (Gell-Mann, 1961; Ne'eman, 1961). This was called the "unitary symmetry" group at the time, and is called the "flavor" SU(3) group today. It was curious that, although the simplest representations of SU(3) have dimensions 3 and 6, only higher representations seemed to occur in nature. The lightest bosonic particles and resonances formed two 8-dimensional representations of SU(3), one consisting of pseudoscalar mesons and the other of vector mesons. The lightest spin $1/2$ baryons also formed an 8-dimensional representation while the lightest spin $3/2$ baryons formed a 10-dimensional representation. The assumption that the interactions that violated unitary symmetry transformed in a specific simple way under the symmetry group led to many relations among particle masses that were accurately obeyed (Coleman and Glashow, 1961; Gell-Mann, 1961; Okubo 1962).

Naturally, the success and utility of unitary symmetry led to searches for further symmetries of the strong interactions, and for models that would give the new abstract symmetries a natural dynamical setting.

An especially appealing generalization of the internal symmetry group SU(3) was the hybrid of internal SU(3) symmetry and intrinsic spin based on the group SU(6) (Sakita, 1964; Gürsey and Radicati, 1964; Pais, 1964; Gürsey et al., 1964). As a method of grouping particles into multiplets it was strikingly successful. The ensemble of pseudoscalar and vector mesons formed a single 35-dimensional representation of SU(6) and the spin $1/2$ and spin $3/2$ baryons formed a single 56-dimensional representation. A similar hybrid symmetry had been successfully used in nuclear physics, but there was an conceptual difference between particle physics and nuclear physics. The theory of nuclei was framed in a non-relativistic context, and nuclear reactions could conserve angular momentum and intrinsic spin separately. However, relativity is inherent in particle physics, and intrinsic spin and orbital angular momentum are not separately conserved. The only conserved rotational quantum number in relativistic quantum mechanics is the total angular momentum.

Soon after SU(6) was proposed, several papers explored the problems associated with formulating SU(6) symmetry, and other hybrid symmetries, in a relativistic context (Coleman, 1964; Michel, 1964; Michel and Sakita, 1964; Bég and Pais, 1964; McGlinn, 1964; Weinberg, 1965; Jordan, 1965). The Coleman-Mandula theorem expressed clearly the reasons that hybrid symmetries could not be invariances of particle physics, and that the only possible Lie groups that can be symmetries of a relativistic particle theory are (locally) isomorphic to the direct product of the Poincaré group and an internal symmetry group.

## Proof of the theorem

As we remarked in the introduction, the Coleman-Mandula theorem rests on the incompatibility of Poincaré invariance and the conservation of hybrid quantum numbers that involve spin. Because the result involves the interplay of relativistic scattering theory and group representation theory, the proof is quite convoluted. The logical structure of the argument is to begin with an arbitrary symmetry group generator and whittle its structure down to the sum of a translation, a pure Lorentz transformation, and an internal symmetry generator.

The basic assumptions about the theory are:

1. The theory is Poincaré invariant, so its symmetry group includes the Poincaré group as a subgroup. In particular, any sum of translations or homogeneous Lorentz transformations of a generator is also in the generator algebra.
2. The number of particle types whose masses are less than any given value is finite. The states of each particle type transform according to some positive mass representation of the Poincaré subgroup of the full symmetry group of the theory. There is no generalization of the Coleman-Mandula theorem to theories with massless particles.
3. The $S$ matrix of the theory has the canonical analyticity properties. For a physical elastic scattering process $p + q \to p' + q'$, the momenta are on shell, ${{p}^{2}}={{{p}'}^{2}}=m_{p}^{2}$ and ${{q}^{2}}={{{q}'}^{2}}=m_{q}^{2}$, and the $T$ matrix, defined as $S - 1$ with the energy-momentum conservation $\delta$ function factored out, is an analytic function of the center-of-mass energy and momentum transfer variables \begin{align}\tag{1} & s={{(p+q)}^{2}} \\ & t={{(p-{p}')}^{2}} \\ \end{align}
4. The theory admits non-trivial scattering. The $T$ matrix does not vanish, except possibly at a discrete set of exceptional values of the center-of mass energy and momentum transfer.
5. An "ugly technical assumption", that the matrix elements of the group generators are distributions in momentum space.

To express these assumptions mathematically requires some notation. We denote the generators of translations (the 4-momenta) by ${{P}_{\mu}}$ and the generators of pure Lorentz transformations by ${M_ {\mu \nu} }$. We will call a general element of the algebra of the generators of the symmetry group $A$, and work with representations in which the 4-momentum is diagonal. On single particle states, the elements of the representation matrices are the matrix elements of $A$ between states of fixed momentum.

$\tag{2}{{A}_{{i}'{\lambda}';i\lambda}}({p}',p)=\left\langle {i}'{\lambda}'{p}'\left| A \right. | i\lambda p \right\rangle$

As before, $i({i}')$, $\lambda ({\lambda}')$, and $p(p')$ denote the initial(final) particle type, spin, and momentum. We will suppress the discrete indices and regard $A(p',p)$ as a matrix valued distribution in the momenta. Two particle states transform under the symmetry group like the direct product of their single particle components. In terms of group generator representations, this is expressed as

$\tag{3}A({p}'{q}',p\,q)=A({p}',p)\otimes I({q}',q)+I({p}',p)\otimes A({q}',q)$

The unit symbol $I({p}',p)$ is ${{\delta}^{4}}({p}'-p)$ times the identity matrix in the discrete indices.

It should be remarked that assumption 5, the "ugly technical assumption" is needed because the apparatus of local field theory is not assumed. In particular, there is no assumption that the generators are integrals of the time components of local currents. That assumption would, of course, imply that all generators are distributions in momentum space. Assumption 5 seems to be the minimum necessary to prove the theorem in the broader context considered here.

The proof of the Coleman-Mandula theorem can be separated into a series of intermediate results as follows:

#### Step 1: Show that $A({p}',p)$ is non-zero only when ${p}'=p$.

To this end we define a momentum projection of $A$ by

$\tag{4}{{A}^{[r]}}=\int{{{d}^{4}}}a\ {{e}^{i{{a}_{\mu}}{{P}^{\mu}}}}\,A\,\,{{e}^{-i{{a}_{\mu}}{{P}^{\mu}}}}\,\hat{r}(a)$

Here $r(\delta)$ is an arbitrary real function of momentum, and $\hat{r}(a)$ is its Fourier transform. Since translations are elements of the full symmetry group, ${{A}^{[r]}}$ is also an element of the group generator algebra. By construction, ${{A}^{[r]}}$ has non-zero matrix elements only between states whose momenta differ by a vector in the support of $r$, since

$\tag{5}{{A}^{[r]}}({p}',p)=r({p}'-p)\ A({p}',p)$

The following trivial observation is the basis of the first part of the proof: If an arbitrary momentum vector $\delta$ is added to an on-shell momentum vector $p$, the result, in general, is not on shell. The values of $\delta$ for which $(p+\delta)$ is on shell lie on a hyperboloid in $\delta$ space parallel to the $p$ mass shell hyperboloid, but centered at $(-p)$ rather than at the origin.

To apply this to a physical elastic scattering process $p+q\to {p}'+{q}'$, with all momenta fixed and on shell, choose the support of $r$ to consist of all momenta $\delta$ such that $(p+\delta)$ is on shell but $(q+\delta)$, $({p}'-\delta)$, and $({q}'-\delta)$ are all off shell. This region is the mass hyperboloid in $\delta$ space ${{(p+\delta)}^{2}}=m_{p}^{2}$, but excluding those points that are also on the ${{(q+\delta)}^{2}}=m_{q}^{2}$, ${{({p}'-\delta)}^{2}}=m_{p}^{2}$, or ${{({q}'-\delta)}^{2}}=m_{q}^{2}$ mass hyperboloids. The excluded points form a dimension 2 subset of the dimension 3 hyperboloid ${{(p+\delta)}^{2}}=m_{p}^{2}$, so they are a subset of measure $0$. The origin in $\delta$ space is always an excluded point, i.e., $r(0)=0$.

Since ${{A}^{[r]}}$ is an element of the symmetry generator algebra, it commutes with the $S$ matrix. The two particle matrix element of the symmetry relation $S{{A}^{[r]}}={{A}^{[r]}}S$ is

$\tag{6}\left\langle {p}'{q}' \right|S\,\left({{A}^{[r]}}\left| p \right\rangle \otimes \left| q \right\rangle +\left| p \right\rangle \otimes {{A}^{[r]}}\left| q \right\rangle \right)=\left(\left\langle {{p}'} \right|{{A}^{[r]}}\otimes \left\langle {{q}'} \right|+\left\langle {{p}'} \right|\otimes \left\langle {{q}'} \right|{{A}^{[r]}} \right)S\,\left| pq \right\rangle$

By construction of ${{A}^{[r]}}$, all terms but the first on the left hand side vanish, leaving

$\tag{7}\left\langle {p}'{q}' \right|S\left({{A}^{[r]}}\left| p \right\rangle \otimes \left| q \right\rangle \right)=0$

There are two possible ways this condition can be satisfied: Either the $S$ matrix is zero or the generator ${{A}^{[r]}}$ annihilates the state $\left|p \right\rangle$. While there is nothing amiss if the $S$ matrix vanishes at some isolated values of the kinematic variables $s$ and $t$, by a suitable choice of $q$, ${p}'$, and ${q}'$ for fixed $p$, the kinematic variables $s$ and $t$ can be given a continuous range of values. The assumed analyticity of the $S$ matrix and the existence of non-trivial scattering excludes the vanishing of $S$ as the explanation. Furthermore, the only value of $\delta$ that is generically excluded from the support of $r$ is the origin. The arbitrariness of $r(\delta)$ gives the conclusion of the first part of the proof, that $A({p}',p)$ vanishes when the momenta are unequal.

$\tag{8}A({p}',p)=0\quad \ \text{for}\;\text{all}\quad \ {p}'\ne p$

#### Step 2: Elucidate the structure of $A({p}',p)$ as a distribution in the momentum variables.

By the technical assumption that any matrix element of a group generator is a distribution in momentum space, it must be of a finite sum of derivatives of ${{\delta}^{4}}({p}'-p)$. The physical requirement that ${p}'$ and $p$ be on shell implies that the derivatives must be along directions tangential to the mass hyperboloid.

We can express these requirements for a generator whose matrix elements have derivatives up to order $N$ explicitly (suppressing the spin and particle type indices of the state vectors) as

$\tag{9}{{A}^{(N)}}({p}',p)=\left\langle {{p}'} \right|{{A}^{(N)}}\left| p \right\rangle =\sum\limits_{n=0}^{N}{{{A}^{(n)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{n}}}}\frac{\partial}{\partial {{p}_{{{\mu}_{1}}}}}}\frac{\partial}{\partial {{p}_{{{\mu}_{2}}}}}\cdot \cdot \cdot \frac{\partial}{\partial {{p}_{{{\mu}_{n}}}}}\ \ {{\delta}^{4}}({p}'-p)$

The factors ${{A}^{(n)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{n}}}}$ are finite matrices whose elements are labeled by the spin and particle type indices. The leading coefficient is isolated by multiple commutation with the momentum operator.

$\tag{10}~{{A}^{(N)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}{{\delta}^{4}}({p}'-p)=\frac{1}{N!}\left\langle {{p}'} \right|\left[ {{P}_{{{\mu}_{1}}}},\left[ {{P}_{{{\mu}_{2}}}},\cdot \cdot \cdot \ \left[ {{P}_{{{\mu}_{N}}}},{{A}^{(N)}} \right] \right] \right]\left| p \right\rangle$

Note that since ${{P}_{\mu}}$ is an element of the symmetry algebra, the $N$-fold commutator of ${{A}^{(N)}}$ with ${{P}_{\mu}}$ is as well. It is an $N=0$ generator, whose matrix elements are given by Eq. (10).

Mathematically, the requirement that the derivatives are in directions tangent to the mass hyperboloid, so that the generators keep $p$ on shell, may be implemented as

$\tag{11}\left[ {{A}^{(N)}},\ {{P}^{\lambda}}{{P}_{\lambda}} \right]\ \ =\ \ 0$

#### Step 3: Isolate the momentum dependence of the $N=0$ representations.

In this step we factor out the energy-momentum $\delta$ function from the symmetry generators and the $S$ matrix, and are left with finite matrix functions of momentum. We use the same symbols, $S({p}'{q}',p\,q)$, ${{A}^{(0)}}(p)$, and $I(p)$, for the finite matrices corresponding to the indicated quantities.

The following argument shows that the trace of the single particle representation matrix of each $N=0$ generator is just a linear function of the momentum variable.

The two particle matrix element of the commutator of $S$ and ${{A}^{(0)}}$, which vanishes because ${{A}^{(0)}}$ is a symmetry generator, gives

\tag{12}\begin{align} & S({p}'{q}',p\,q)\left({{A}^{(0)}}(p)\ \otimes I(q)+I(p)\ \otimes {{A}^{(0)}}(q) \right) \\ & \quad \quad \quad =\left({{A}^{(0)}}({p}')\ \otimes I({q}')+I({p}')\ \otimes {{A}^{(0)}}({q}') \right)S({p}'{q}',p\,q) \\ \end{align}

Since the $S$ matrix is unitary, the representation matrices on each side of this equality are similarity transforms of each other, and so their traces are equal. This gives the remarkable functional relation

$\tag{13}\frac{Tr\,{{A}^{(0)}}({p}')}{N({{m}_{p}})}+\frac{Tr\,{{A}^{(0)}}({q}')}{N({{m}_{q}})}\quad =\quad \frac{Tr\,{{A}^{(0)}}(p)}{N({{m}_{p}})}+\frac{Tr\,{{A}^{(0)}}(q)}{N({{m}_{q}})}$

Here $N({{m}_{p}})=Tr\,I(p)$ is the number of states in the $p$ representation, which is finite by assumption 2. Since there is only one constraint among the momenta, ${p}'+{q}'=p+q$, this relation can only be satisfied if each term is the same linear function of its argument,

$\tag{14}Tr\ {{A}^{(0)}}(p)=\left({{a}_{\mu}}{{p}^{\mu}}+b \right)N({{m}_{p}})$

The coefficients ${{a}_{\mu}}$ and $b$ depend only on the element of the generator algebra. They are independent of the representations according to which the particles transform and of their momenta.

The distribution structure of symmetry generator representations with $N=0$ is

$\tag{15}{{A}^{(0)}}({p}',p)\ \ =\ \ {{A}^{(0)}}(p)\ {{\delta}^{4}}({p}'-p)$

Since there are no derivatives acting on the $\delta$ function, matrix elements of the commutator of ${{A}^{(0)}}$ with the momentum operator ${{P}_{\mu}}$ have a factor of $({{{p}_{\mu}}'}-{{p}_{\mu}})$ evaluated at ${p}'=p$. Thus all elements of the symmetry generator algebra whose representations have $N=0$ commute with the 4-momentum.

Since $Tr\,{{A}^{(0)}}(p)$ is the same for all representations, we can usefully define reduced symmetry generators $B$ by separating out the momentum and the constant term from $Tr\,{{A}^{(0)}}(p)$.

$\tag{16}{{A}^{(0)}}\ \,=\ \,{{a}_{\mu}}{{P}^{\mu}}+bI+\ B$

The numerical constants ${{a}_{\mu}}$ and $b$ are as in Eq. (14), and depend only on the generator ${{A}^{(0)}}$, and $I$ is the identity operator. The $B$ terms are themselves $N=0$ elements of the symmetry generator algebra. The traces of the coefficient matrices $B(p)$ multiplying the energy-momentum $\delta$ function vanish by construction. Furthermore, because the momenta commute with the $B$ generators and the trace of the commutator of any pair of finite matrices is zero, they close under commutation, and form a subalgebra of the algebra of $N=0$ symmetry generators.

#### Step 4: Show that the reduced $N=0$ operators generate only internal symmetries.

This argument uses the fact that $N=0$ generators commute with the momentum operators to show that they also commute with Lorentz transformations and do not act on spin indices, which is the definition of an internal symmetry. The following demonstration is much clearer than that given in the Coleman-Mandula paper. It is adapted from Volume III of Steven Weinberg's Quantum Field Theory treatise (Weinberg, 2000).

Denote by ${{B}_{\alpha}}$ a finite set of Hermitean generators that spans the reduced $N=0$ symmetry subalgebra. Their commutators are given by the real structure constants $C_{\alpha \beta}^{\gamma}$

$\tag{17}\left[{{B}_{\alpha}},\ {{B}_{\beta}}\right]=i\ C_{\alpha \beta}^{\gamma}{{B}_{\gamma}}$

Denote a finite Lorentz transformation by $\Lambda$ and the operator that effects this transformation on Hilbert space by $U(\Lambda)$. The Lorentz transforms of the generators, $U(\Lambda){{B}_{\alpha}}{{U}^{\mathsf{\dagger}}}(\Lambda)$, are also Hermitean reduced $N=0$ generators and also commute with ${{P}^{\mu}}$. Since we have taken the ${{B}_{\alpha}}$ to be a complete set of generators that span the reduced $N=0$ subalgebra, the transformed generators must be real linear combinations of the ${{B}_{\alpha}}$.

$\tag{18}U(\Lambda){{B}_{\alpha}}{{U}^{\mathsf{\dagger}}}(\Lambda)\ \ =\ \ D_{\alpha}^{{{\alpha}'}}(\Lambda)\,{{B}_{{{\alpha}'}}}$

The $D_{\alpha}^{{{\alpha}'}}(\Lambda)$ matrices form a real representation of the homogeneous Lorentz group. From the fact that the commutation relations of the similarity transformed generators have the same structure constants as the ${{B}_{\alpha}}$, there follows an invariance relation on the 3 index structure constant tensor

$\tag{19}C_{\alpha \beta}^{\gamma}\ \ =\ \ D_{\alpha}^{{{\alpha}'}}(\Lambda)\,D_{\beta}^{{{\beta}'}}(\Lambda)\,D_{{{\gamma}'}}^{\gamma}({{\Lambda}^{-1}})\,\ C_{{\alpha}'{\beta}'}^{{{\gamma}'}}$

and a similar relation follows for the Lie algebra metric $\ {{g}_{\alpha \beta}}\ =\ C_{\alpha \delta}^{\gamma}\,\ C_{\gamma \beta}^{\delta}$

$\tag{20}{{g}_{\alpha \beta}}\ \ =\ \ D_{\alpha}^{{{\alpha}'}}(\Lambda)\,\,D_{\beta}^{{{\beta}'}}(\Lambda)\,\,{{g}_{{\alpha}'{\beta}'}}$

We now reserve the indices $\mu$ and $\nu$ for the momenta and other Greek indices for all other quantum numbers. Since all $N=0$ generators commute with the momenta, the associated structure constants $C_{\mu \beta}^{\gamma}$ and $C_{\alpha \mu}^{\gamma}$vanish, as do ${{g}_{\mu \beta}}$ and ${{g}_{\alpha \mu}}$. This shows that the metric has a block structure: the elements that have an energy-momentum index, ${{g}_{\alpha \nu}}$, ${{g}_{\nu \beta}}$, and ${{g}_{\mu \nu}}$, are all $0$, while the remaining submatrix ${{g}_{\alpha \beta}}$ is positive definite and symmetric. Restricting ourselves to the ${{g}_{\alpha \beta}}$ submatrix, the invariance relation Eq. (20) can be rewritten as

$\tag{21}\left({{g}^{-1/2}}\,D\,{{g}^{1/2}} \right)\ \ {{\left({{g}^{-1/2}}\,D\,{{g}^{1/2}} \right)}^{\Tau}}=\ \ 1$

This shows that ${{g}^{-1/2}}\,D(\Lambda)\,\,{{g}^{1/2}}$ is a real orthogonal unitary representation of the homogeneous Lorentz group. However, because the Lorentz group is non-compact, it has no non-trivial finite dimensional unitary representations.

We arrived at this "contradiction" by making the tacit assumption that the representation $D(\Lambda)$ in Eq. (18) was non-trivial. In fact, we have just inferred that it must be the identity representation $D(\Lambda)=1$. Therefore, the entire algebra of $N=0$ generators ${{B}_{\alpha}}$ commutes with Lorentz transformations.

Consider the action of ${{B}_{\alpha}}$ on a single particle state with momentum $p$ and take the Lorentz transformation $\Lambda$ to be an arbitrary element of the little group of $p$, the subgroup of Lorentz transformations that leave the momentum $p$ of the state unchanged. The ensemble of $U(\Lambda)$ transformations effect arbitrary rotations of the spin of the state. The fact that all reduced $N=0$ generators ${{B}_{\alpha}}$ commute with all $U(\Lambda)$, Eq. (18) with $D(\Lambda)=1$, means that that they act trivially on the spin degrees of freedom. To sum up, the $N=0$ sector of the symmetry algebra is the direct sum of the algebra of the generators of translations, spanned by the 4-momenta ${{P}_{\mu}}$, and the algebra of internal symmetries, which are momentum independent and do not act on the spin indices.

#### Step 5: Show that there are no symmetry generators with $N>1$ derivatives.

To show that there are no generators with more than 1 derivative, consider the construction of ${{A}^{(N)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}{{\delta}^{4}}({p}'-p)$ in Eq. (10). It is the single particle matrix element of an $N=0$ symmetry generator. From Step 3, this means that the trace of ${{A}^{(N)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ is a linear function of the momentum

$\tag{22}Tr\ {{A}^{(N)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}={{a}_{\nu {{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}{{p}^{\nu}}+{{b}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$

and Eq. (9) shows that ${{A}^{(N)}}{{(p)}_{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ is completely symmetric in its indices, and so the ${{a}_{\nu {{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ coefficients are symmetric in the second through ${{(N+1)}^{\mathsf{st}}}$ indices $\{{{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}\}$.

From Eqs. (10) and (14), the matrix trace of the multiple commutator of ${{P}^{\lambda}}{{P}_{\lambda}}$ and $N-1$ factors of ${{P}_{{{\mu}_{i}}}}$ with ${{A}^{(N)}}$ is

\tag{23}\begin{align} & \frac{1}{N!}Tr\ \left\langle {p}'\left| \left[ {{P}^{\lambda}}{{P}_{\lambda}},\,\,\left[ {{P}_{{{\mu}_{2}}}},\cdot \cdot \cdot \ \left[ {{P}_{{{\mu}_{N}}}},{{A}^{(N)}} \right] \right] \right] \right|p \right\rangle \\ & \quad \quad \quad \quad \quad = 2 {{p}^{\lambda}}\left({{a}_{\nu \lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}{{p}^{\nu}}+{{b}_{\lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}} \right)\ {{\delta}^{4}}({p}'-p)\end{align}

This must vanish for any value of $p$ because it is the commutator of ${{P}^{\lambda}}{{P}_{\lambda}}$ with an element of the symmetry algebra, which keeps states on shell. Thus the ${{b}_{\lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ coefficients are zero and the ${{a}_{\nu \lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ coefficients must be antisymmetric in the $\lambda$ and $\nu$ indices. However, the latter is impossible because it is in contradiction with the complete symmetry of ${{a}_{\nu \lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ in the $\{\lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}\}$ indices. Explicitly,

\tag{24}\begin{align} {{a}_{\nu \lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}} & =-\ {{a}_{\lambda \nu {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}=-\,{{a}_{\lambda {{\mu}_{2}}\nu \cdot \cdot \cdot {{\mu}_{N}}}} = +\,{{a}_{{{\mu}_{2}}\lambda \nu \cdot \cdot \cdot {{\mu}_{N}}}} \\ & \quad \quad = +\,{{a}_{{{\mu}_{2}}\nu \lambda \cdot \cdot \cdot {{\mu}_{N}}}}=-\,{{a}_{\nu {{\mu}_{2}}\lambda \cdot \cdot \cdot {{\mu}_{N}}}}=-\,{{a}_{\nu \lambda {{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}\end{align}

Thus there are no symmetry generators with more than one momentum derivative acting on the momentum conservation $\delta$ function.

Lest the reader fear that we have excluded all generators, note that the above argument required that the coefficient ${{a}_{\nu {{\mu}_{1}}{{\mu}_{2}}\cdot \cdot \cdot {{\mu}_{N}}}}$ have at least three indices. That is, the foregoing says nothing about the cases $N=0$ or $N=1$.

#### Step 6: Identify the Poincaré transformation generators.

The $N=1$ generators have the form

$\tag{25}{{A}^{(1)}}({p}',p)=\left(\left(a_{\nu \mu}^{(1)}{{p}^{\nu}}+b_{\mu}^{(1)} \right)\frac{\partial}{\partial {{p}_{\mu}}}+A_{subleading}^{(1)}(p) \right)\ \ {{\delta}^{4}}({p}'-p)$

As shown in the previous Step, the coefficient $a_{\nu \mu}^{(1)}$ is antisymmetric in its indices and the coefficient $b_{\mu}^{(1)}$ vanishes. The first term, $\ a_{\nu \mu}^{(1)}{{p}^{\nu}}(\partial /\partial {{p}_{\mu}}){{\delta}^{4}}({p}'-p)$, is just the geometric part of a Lorentz generator. If we subtract the full Lorentz transformation generator from ${{A}^{(1)}}({p}',\ p)$

$\tag{26}{{A}^{(1)}}({p}',p)\ \ -\ \ a_{\nu \mu}^{(1)}\,{{M}^{\nu \mu}}=\ \ {{{A}'}^{(0)}}\ {{\delta}^{4}}({p}'-p)$

we obtain yet another $N=0$ element of the generator algebra. The part of the Lorentz generator $a_{\nu \mu}^{(1)}\,{{M}^{\nu \mu}}$ that acts on the spin indices is included in ${{{A}'}^{(0)}}$. However, in Step 4 we showed that any $N=0$ generator was the sum of a translation generator and an internal symmetry generator. Even though we have nominally incorporated the spin part of $a_{\nu \mu}^{(1)}\,{{M}^{\nu \mu}}$, into ${{{A}'}^{(0)}}$, the result of the analysis in Step 4 ensures that the operator so obtained does not affect the spin indices.

This completes the demonstration that the most general symmetry algebra of a Poincaré invariant theory with non-trivial scattering is a direct sum of the Lorentz algebra $(N=1)$ generated by the angular momentum and boost generators ${{M}_{\mu \nu}}$ , the algebra of translation generators $(N=0)$ generated by the linear momentum ${{P}_{\nu}}$, and an algebra of pure internal symmetry transformation generators $(N=0)$.

## Necessity of all the assumptions

The physics assumptions used in the proof of the Coleman-Mandula theorem are relativistic invariance and the existence of scattering. Both are needed. What fails in trying to implement a hybrid symmetry is not the possibility of grouping particles into representations of hybrid symmetry, but rather the conservation of the hybrid quantum numbers in scattering reactions or particle decays. To make the need for both assumptions clear, let us review examples of models that do display hybrid symmetries.

There is a well-known approximate hybrid symmetry in nuclear physics. It is the spin-isospin symmetry in the shell model introduced by Wigner (Wigner, 1937). The symmetry is represented on single nucleon states by arbitrary unitary mixing of the four basis states $\left| p\uparrow \right\rangle$, $\left| p\downarrow \right\rangle$, $\left| n\uparrow \right\rangle$, and $\left| n\downarrow \right\rangle$. The fact that in the real world this SU(4) symmetry group holds only approximately is not the point. It is a consistent model for the excitation level spectrum and, more relevantly, for nuclear decays. This existence of this example shows that there can be no analogue of the Coleman-Mandula theorem for non-relativistic systems.

There are also examples of relativistic theories with exact hybrid symmetries that only apply to the classification of bound states but not to the scattering of the bound states, but they are far less familiar (and less useful) than nuclear spin-isospin symmetry. The O(4) symmetry of the hydrogen atom carries over to the relativistic version of the same bound state problem: the Bethe-Salpeter equation for two massive scalar particles bound by the exchange of a massless scalar. Cutkosky solved this problem by Wick rotating the momenta to Euclidean space and mapping the relative momentum to the surface of a 4-sphere. The resulting equation was invariant under rotations on the 4-sphere, and so had the same O(4) symmetry as the non-relativistic hydrogen atom (Cutkosky, 1954).

In Minkowski space one may follow the same procedure, but there is a subtlety in identifying the physical symmetry transformations. It is necessary to enlarge the bound state symmetry group to O(4,C), the complex generalization of O(4). In any frame there is an O(4) subgroup of O(4,C) that is represented unitarily on the bound state wave functions. However, in each frame this physical O(4) subgroup is different. This structure is a reflection of the fact that the Lorentz group has no finite dimensional unitary representations. When each of the six generators of the physical O(4) subgroup in a given frame is boosted, it becomes a complex linear combination of the original set. The original set of six generators is thus effectively augmented by six new ones, each being $i$ times one of the original set. This set of twelve generators is closed under Lorentz transformations, showing that the full bound state symmetry group is O(4,C) (Mandula, 1969).

## Supersymmetry and the Haag-Łopuszański-Sohnius theorem

The Coleman-Mandula theorem deals only with symmetries expressed in terms of Lie groups, whose structure is described by the commutation relations between their generators. There are symmetries that cannot be so expressed, however. A class of such symmetries, called supersymmetries, involve transformations that change bosons into fermions and vice versa. These symmetries were discovered several years after the Coleman-Mandula theorem was proved.

Supersymmetries were first discovered in the context of string theory. Gervais and Sakita formulated an action for a theory with fermionic as well as bosonic variables, and observed that their action was invariant under set of transformations that interchanged the fermionic and bosonic world sheet fields (Gervais and Sakita, 1971). This was effectively a supersymmetry of a two dimensional field theory. A couple of years later, Wess and Zumino succeeded in extending the idea to four dimensional field theory (Wess and Zumino, 1974a). In a subsequent paper, they traced the reason that the Coleman-Mandula theorem does not apply to supersymmetries to the fact that the generators of supersymmetries are fermionic operators, and their structures are expressed by anticommutation relations (Wess and Zumino, 1974b).

Nonetheless, the possible supersymmetries are quite as restricted as ordinary symmetries. The restrictions on the possible supersymmetries were found by Haag, Łopuszański, and Sohnius, and is given by the theorem that bears their names (Haag et al., 1975). The proof of the Haag-Łopuszański-Sohnius theorem follows the same strategy as that of the Coleman-Mandula theorem. That is one begins with a completely general supersymmetry generator and, step by step, finds restrictions on the allowed generators and their anticommutators.

Unlike internal symmetries, supersymmetries are not Lorentz scalars, and their Lorentz transformation properties are relevant. Denote the generators of the commuting SU(2) factors of the Lorentz group by $J_{i}^{\pm}$ and denote a generic supersymmetry generator transforming according to the $({{j}^{+}},\,{{j}^{-}})$ representation of the Lorentz group by ${{Q}_{ar}}$. The index $a$ labels the Lorentz component $({{m}^{+}},\,\,{{m}^{-}})$ and the index $r$ distinguishes different generators with the same Lorentz transformation properties. The Hermitean adjoint operator $Q_{ar}^{\mathsf{\dagger}}$ transforms according to the conjugate representation $({{j}^{-}},\,{{j}^{+}})$.

In terms of the foregoing, the conclusions of the Haag-Łopuszański-Sohnius theorem are as follows:

1. The only possible fermionic supersymmetry generators ${{Q}_{ar}}$transform like the $(1/2, 0)$ representation and their conjugates ${{\bar{Q}}_{ar}}$ transform like $(0,1/2)$ in the standard basis. The linear relation between the conjugates ${{\bar{Q}}_{ar}}$ and the Hermitean adjoints of the ${{Q}_{ar}}$ is $\tag{27}{{\bar{Q}}_{ar}}=\ \ {{e}_{a{a}'}}\,Q_{{a}'r}^{\,\mathsf{\dagger}}$ where $e$ is an antisymmetric $2 \times 2$ matrix. The index $a$ takes the values $\pm 1/2$, and we fix the phase of $e$ by choosing ${\,{e}_{+{\scriptscriptstyle {}^1\!/\!{}_2}\,-{\scriptscriptstyle {}^1\!/\!{}_2}}}=+1$. The index $r$ distinguishes between different multiplets of supercharges.
2. The supercharges can be chosen so that their anticommutation relations are \tag{28}\begin{align} & \left\{{{Q}_{ar}},Q_{bs}^{\,\mathsf{\dagger}}\right\}=2\,{{\delta}_{rs}}\,\sigma _{ab}^{\,\mu}\,{{P}_{\mu}} \\ & \left\{{{Q}_{ar}},Q_{bs}^{\phantom{\,\mathsf{\dagger}}}\right\}={{e}_{ab}}\,{{Z}_{rs}} \\ \end{align} The matrices ${{\sigma}^{\,i}}\ (i=1,\,2,\,3)$ are the Pauli spin matrices, ${{\sigma}^{\,0}}$ is the $2 \times 2$ unit matrix, and $e$ is as above.
3. The supersymmetry generators ${{Q}_{ar}}$ and $Q_{as}^{\,\mathsf{\dagger}}$ commute with the translation generators ${{P}_{\mu}}$. The bosonic operators ${{Z}_{rs}}$ are antisymmetric in their indices and are "central charges" of the symmetry algebra, that is, they are symmetry generators that commute with all elements of the symmetry algebra. Therefore, the ${{Z}_{rs}}$ generators are singlets under Lorentz transformations and under the internal symmetry group.

Like the proof of the Coleman-Mandula Theorem, the proof of the Haag-Łopuszański-Sohnius theorem proceeds in several steps, as follows:

#### Step 1: Determine the Lorentz transformation properties of the supercharges.

Supersymmetry generators are fermionic charges, so they transform according to representations $({{j}^{+}},\,{{j}^{-}})$, where one of the labels ${{j}^{\,\pm}}$ is an integer and the other a half integer. In the usual basis, where ${{m}^{\pm}}$ label the elements of the representation by their eigenvalues of $J_{3}^{\,\pm}$, the adjoint of the supercharge $Q_{({{m}^{+}},\,{{m}^{-}})}^{({{\,j}^{\,+}},\,{{j}^{\,-}})}$ transforms exactly like the $(-{{m}^{-}},-{{m}^{+}})$ element of the $({{j}^{-}},\,{{j}^{+}})$ representation. We denote it by

$\tag{29}\bar{Q}_{(-{{m}^{\,-}},\,-{{m}^{\,+}})}^{\,({\,\,{j}^{\,-}}\,,\,\,{\,\,{j}^{\,+}})}=\ \ {{\left(Q_{({{m}^{\,+}},\,{{m}^{\,-}})}^{(\,{{j}^{\,+}},\,\,{{j}^{\,-}})} \right)}^{\mathsf{\dagger}}}$

From the rule for addition of ordinary angular momentum, one can show that the anticommutator of an extremal element of $Q_{({{m}^{\,+}},\,{{m}^{\,-}})}^{(\,{{j}^{\,+}},\,\,{{j}^{\,-}})}$ with its adjoint transforms as indicated by

$\tag{30}\left\{ Q_{({{j}^{\,+}},\,{-{j}^{\,-}})}^{(\,{{j}^{\,+}},\,\,{{j}^{\,-}})},\,\,\bar{Q}_{({{j}^{-}},\,{-{j}^{\,+}})}^{(\,{{j}^{\,-}},\,\,{{j}^{\,+}})} \right\}=B_{({{j}^{\,+}}+{{j}^{\,-}},\ -{{j}^{\,+}}-{{j}^{\,-}})}^{({{j}^{\,+}}+{{j}^{\ -}},\ \,{{j}^{\,+}}+\,{{j}^{\,-}})}$

The operator $B$ is an ordinary (bosonic) symmetry generator, since both of its representation labels are positive half integers. Furthermore, $B$ cannot vanish, because the anticommutator of any operator with its adjoint can only vanish if the operator vanishes itself. However, the Coleman-Mandula theorem shows that the only bosonic elements of the symmetry algebra are internal symmetries which are Lorentz scalars, the 4-momentum ${{P}_{\mu}}$, transforming like $(1/2, 1/2)$, and the generators of pure Lorentz transformations ${{M}_{\mu \nu}}$, which transform according to the reducible representation $(1,0) \oplus (0,1)$. The only solution to these constraints on the transformation properties of $B$ is that ${\,{j}^{+}}\!+{{j}^{-}}$ must be $1/2$, and so $B$ is proportional to the 4-momentum. Therefore, one of ${\,{j}^{\pm}}$ is $1/2$ and the other is $0$, and we conclude that the only allowed fermionic generators must transform like $(1/2, 0)$ or $(0, 1/2)$.

#### Step 2: Verify that the $\{{{Q}_{ar}},Q_{bs}^{\,\mathsf{\dagger}}\}$ anticommutator satisfies the first equality of Eq. (28).

The first anticommutator in Eq. (28) above follows from the Coleman-Mandula theorem: Since ${{P}_{\mu}}$ is the only allowed $(1/2, 1/2)$ symmetry generator, and the ${{\sigma}^{\,\mu}}$ matrices are the coefficients expressing the elements of a 4-vector in bi-spinor notation, the form of that anticommutator must be

$\tag{31}\left\{ {{Q}_{ar}},Q_{bs}^{\,\mathsf{\dagger}} \right\}\ \ =\ \ 2\,{{C}_{rs}}\sigma _{ab}^{\,\mu}\,{{P}_{\mu}}\ \ \equiv \ \ 2\,{{C}_{rs}}{{P}_{ab}}$

The matrix $C$ is positive definite, as can be seen as follows: Take $a=b$; the absence of massless particles implies that the diagonal elements of ${{P}_{ab}}$ are both positive. Contract the $r$ and $s$ indices above with an arbitrary row vector and its conjugate column vector. This gives the anticommutator of a supercharge and its adjoint, which must be positive. Since $C$ is positive definite, the given supercharges may be replaced by the real linear combinations given by the inverse square root of $C$. These new supercharges satisfy the first anticommutator of Eq. (28).

#### Step 3: Show that the supercharges commute with 4-momenta.

The argument that the supercharges commute with 4-momentum explicitly uses the transformation properties of the supercharges. The commutator of the momentum operator, transforming like $(1/2, 1/2)$, with a supercharge, transforming like $(1/2, 0)$, must be proportional to a conjugate supercharge, transforming like $(0,1/2)$, because $(0,1/2) \otimes (1/2, 1/2) = (1/2,0) \oplus (1/2,1)$, and there are no supersymmetry generators transforming like $(1/2,1)$. The detailed form of the relation is fixed by the Lorentz transformation properties of the supercharges and the 4-momentum:

\tag{32}\begin{align} \left[{{P}_{ab}},\,{{Q}_{cr}^{\phantom{\mathsf{\dagger}}}}\right] & = +{{C}_{rs}}\,{{e}_{ac}}\,Q_{bs}^{\mathsf{\dagger}} \\ \left[{{P}_{ab}},\,Q_{cs}^{\mathsf{\dagger}}\right] & = -C_{sr}^{*}\,{{e}_{bc}}\,{{Q}_{ar}} \\ \end{align}

Using the above it is straightforward to compute the triple commutator-anticommutator

$\tag{33}\left[ {{P}_{ab}},\,\left[ {{P}_{cd}},\left\{ {{Q}_{er}},\,Q_{fs}^{\mathsf{\dagger}} \right\} \right] \right]\ \ =\,\ 2\,{{\delta}_{rs}}\left[ {{P}_{ab}},\,\left[ {{P}_{cd}},\,{{P}_{ef}} \right] \right]$

The right hand side is zero for any values of the indices because translations form an Abelian group. However, the left hand side is not identically zero. To pick one example, Eq. (32) gives

$\tag{34}\sum\limits_{r}{\ \left[ {{P}_{+{\scriptscriptstyle {}^1\!/\!{}_2}\,+{\scriptscriptstyle {}^1\!/\!{}_2}}}\,,\,\left[ {{P}_{+{\scriptscriptstyle {}^1\!/\!{}_2}\,+{\scriptscriptstyle {}^1\!/\!{}_2}}}\,,\left\{ {{Q}_{-{\scriptscriptstyle {}^1\!/\!{}_2}\,r}},\,Q_{-{\scriptscriptstyle {}^1\!/\!{}_2}\,r}^{\mathsf{\dagger}} \right\} \right] \right]}\ \ =\,\ -4\left(Tr{{C}^{\mathsf{\dagger}}}C\right)\,{{P}_{+{\scriptscriptstyle {}^1\!/\!{}_2}\,+{\scriptscriptstyle {}^1\!/\!{}_2}}}$

These the two evaluations agree only if $Tr{\,{C}^{\mathsf{\dagger}}}C$, and hence $C$, vanishes. Thus we conclude that ${{P}_{\mu}}$ commutes with ${{Q}_{cr}}$ (and $Q_{cs}^{\mathsf{\dagger}}$).

$\tag{35}\left[{{P}_{ab}},\,\,{{Q}_{cr}^{\phantom{\mathsf{\dagger}}}}\right]\ \ =\ \,\left[{{P}_{ab}},\,Q_{cs}^{\mathsf{\dagger}}\, \right]\ \ =\,\ 0$

#### Step 4: Show that the $Z$ generators are central charges.

Note that the $Z$ operators commute with the 4-momenta because, per Eq. (28), they are comprised of two supercharges, and we have just shown that supercharges commute with the 4-momenta.

$\tag{36}\left[{{P}_{ab}},\,{{Z}_{rs}^{\phantom{\mathsf{\dagger}}}} \right]=\frac{1}{2}{{e}_{cd}}\,\left[ {{P}_{ab}},\left\{ {{Q}_{cr}},\,{{Q}_{ds}^{\phantom{\mathsf{\dagger}}}} \right\} \right]=0$

To show that the $Z$ operators commute with the $Q_{bs}^{\mathsf{\dagger}}$ supercharges, we use their definitions per Eq. (28) and evaluate

$\tag{37}\left[Q_{ct}^{\mathsf{\dagger}},\,\,{{Z}_{rs}} \right]=\frac{1}{2}{{e}_{ab}}\left[ Q_{ct}^{\mathsf{\dagger}},\left\{ {{Q}_{ar}^{\phantom{\mathsf{\dagger}}}},{{Q}_{bs}} \right\} \right]={{e}_{ab}}\left({{\delta}_{tr}}\left[ {{P}_{ac}},\,{{Q}_{bs}^{\phantom{\mathsf{\dagger}}}} \right]+{{\delta}_{ts}}\left[ {{P}_{bc}},{{Q}_{ar}^{\phantom{\mathsf{\dagger}}}} \right] \right)$

Both terms on the right vanish, again because the 4-momenta commute with all supercharges. Thus the $Z$ charges commute with the conjugate supercharges.

$\tag{38}\left[Q_{ct}^{\mathsf{\dagger}},\,\,{{Z}_{rs}} \right]=0$

To show that the $Z$ operators commute with the ${{Q}_{a\,r}}$ supercharges, expand the multiple commutator-anticommutator

$\tag{39}\left\{Q_{ct}^{\mathsf{\dagger}},\left[ {{Q}_{au}^{\phantom{\mathsf{\dagger}}}},\,\,{{Z}_{rs}} \right] \right\}=\left[ \left\{ Q_{ct}^{\mathsf{\dagger}},{{Q}_{au}} \right\},{{Z}_{rs}} \right]-\left\{ \left[ Q_{ct}^{\mathsf{\dagger}},{{Z}_{rs}} \right],{{Q}_{au}} \right\}$

Again, both terms on the right vanish. The first is the commutator of a momentum operator with a $Z$ operator (see Eq. (31)), which vanishes, and the second includes the commutator of a $Z$ operator with an adjoint supercharge, which we have just seen vanishes.

Since ${{Z}_{rs}}$ is the sum of products of two generators transforming according to the $(1/2, 0)$ representation of the Lorentz group, it must transform according to the $(0,0)$ or the $(1,0)$ representation, or a linear combination thereof. However, the Coleman-Mandula theorem shows that the only generators of the bosonic symmetry algebra that transform like $(1,0)$ are the Lorentz generators themselves, which do not commute with the 4-momenta. Therefore the $Z$ operators are Lorentz scalars and their commutators with the ${{Q}_{ar}}$ supercharges must transform according to the $(1/2, 0)$ representation. Since the only such symmetry generators are the supercharges themselves, the commutator must be a linear combination of those supercharges

$\tag{40}\left[{{Q}_{au}^{\phantom{\mathsf{\dagger}}}},\,\,{{Z}_{rs}} \right]=C_{uv}^{(rs)}{{Q}_{av}}$

and so the multiple commutator is

$\tag{41}\left\{Q_{ct}^{\mathsf{\dagger}},\left[ {{Q}_{au}^{\phantom{\mathsf{\dagger}}}},\,\,{{Z}_{rs}} \right] \right\}=\left\{ Q_{ct}^{\mathsf{\dagger}},C_{uv}^{(rs)}\,{{Q}_{av}} \right\}=2\,C_{ut}^{(rs)}{{P}_{ac}}\,$

We have seen above that this multiple commutator vanishes, which can only be true if the coefficients in the above linear combination, $C_{ut}^{(rs)}$, are all zero. This implies that the commutators of the ${{Z}_{rs}}$ operators with the ${{Q}_{at}}$ supercharges vanish.

To summarize the analysis so far, including the Hermitean conjugates of the foregoing, we have shown

\tag{42}\begin{align} \left[{{Q}_{at}^{\phantom{\mathsf{\dagger}}}},\,\,{{Z}_{rs}}\right] & = \left[{{Q}_{at}},\,\,Z_{rs}^{\mathsf{\dagger}}\right]=\left[Q_{bt}^{\mathsf{\dagger}},\,\,{{Z}_{rs}}\right]=\left[Q_{bt}^{\mathsf{\dagger}},\,\,Z_{rs}^{\mathsf{\dagger}}\right] \\ & = \left[{{Z}_{rs}^{\phantom{\mathsf{\dagger}}}},\,\,{{Z}_{tu}}\right]=\left[{{Z}_{rs}},\,\,Z_{tu}^{\mathsf{\dagger}}\right]=\left[Z_{rs}^{\mathsf{\dagger}},\,\,Z_{tu}^{\mathsf{\dagger}}\right]=0 \\ \end{align}

This shows that the $Z$ operators are central charges of the pure supersymmetry algebra.

Next consider the commutators of arbitrary bosonic internal symmetry generators ${{B}_{t}}$ with the supercharges and the $Z$ operators formed from them. The supercharges form a representation of the internal symmetry group, i.e.,

$\tag{43}\left[{{B}_{t}},\,\,{{Q}_{ar}^{\phantom{\mathsf{\dagger}}}}\right]\ \ =\ \ D_{r{r}'}^{\,t}\,{{Q}_{a{r}'}}$

where the matrix $D^{\,t}$ is a representation of ${{B}_{t}}$. Therefore the commutator of an internal symmetry with a $Z$ operator is

$\tag{44}\left[{{B}_{t}},\,\,{{Z}_{rs}^{\phantom{\mathsf{\dagger}}}}\right]\ \ =\ \ \left(D_{r{r}'}^{\,t}\,{{\delta}_{s{s}'}}+{{\delta}_{r{r}'}}D_{s{s}'}^{\,t}\right)\,{{Z}_{{r}'{s}'}}$

The $\,{{Z}_{rs}}$ charges are ordinary bosonic symmetries, and Eq. (44) says that they comprise an invariant subalgebra of the full internal symmetry algebra. The $\,{{Z}_{rs}}$ charges all commute with each other, so it is an Abelian invariant subalgebra. Since the $\,{{Z}_{rs}}$ charges are a subset of the ${{B}_{t}}$ charges, the coefficients in Eq. (44) are a subset of the structure constants of the internal symmetry group. If we lower the $t$ index with the group metric, we can write the structure constants in the standard form ${{f}_{t,rs,{r}'{s}'}}$. The index $t$ runs over the all the generators of the internal symmetry algebra while the oddly labeled indices $rs$ and ${r}'{s}'$ only run over the elements of the Abelian invariant subalgebra. The 3-index structure constant array is antisymmetric under exchange of any pair of indices, and the fact that the $\,{{Z}_{rs}}$ operators commute with each other means that the structure constants ${{f}_{rs,{r}'{s}',t}}$ are zero for all values of the indices $rs$, ${r}'{s}'$, and $t$. The antisymmetry of $f$ then implies that the ${{f}_{t,rs,{r}'{s}'}}$, which are the coefficients in Eq. (44), vanish. This shows that the $\,{{Z}_{rs}}$ charges are central charges of the full symmetry algebra, and so completes the proof of the Haag-Łopuszański-Sohnius theorem.

## References

• M. A. B. Bég and A. Pais, Covariance, SU(6), and Unitarity, Phys. Rev. Lett. 14, 509 (1965).
• S. Coleman, Trouble with Relativistic SU(6), Phys. Rev. 138, B1262 (1965).
• S. Coleman and S. L. Glashow, Electrodynamic properties of baryons in the unitary symmetry scheme, Phys. Rev. Lett. 6, 423 (1961).
• S. Coleman and J. Mandula, All Possible Symmetries of the S Matrix, Phys. Rev. 159, 1251 (1967).
• R. E. Cutkosky, Solutions of a Bethe-Salpeter Equation, Phys. Rev. 96, 1135 (1954).
• M. Gell-Mann, The Interpretation of the New Particles as Displaced Charged Multiplets, Il Nuovo Cimento 4, 848 (1956).
• M. Gell-Mann, The Eightfold Way: A Theory of Strong Interaction Symmetry, Caltech Synchrotron Laboratory Report CSTL-20 (1961); Reprinted in "The Eightfold Way", M. Gell-Mann and Y. Ne’eman (W. A. Benjamin, New York, 1964).
• J.-L. Gervais and B. Sakita, Field Theory Interpretation of Supergauges in Dual Models, Nucl. Phys. B, 34, 643 (1971).
• F. Gürsey and L. Radicati, Spin and Unitary Spin Independence of Strong Interactions, Phys. Rev. Lett. 13, 173 (1964).
• F. Gürsey, A. Pais, and L. Radicati, Spin and Unitary Spin Independence of Strong Interactions, Phys. Rev. Lett. 13, 299 (1964).
• R. Haag, J. Łopuszański, and M. Sohnius, All Possible Generators of Supersymmetries of the S Matrix, Nucl. Phys. B, 88, 257 (1975).
• T. F. Jordan, Conservation Laws Implied by Lorentz Invariance and Conservation of Spin, Phys. Rev. 140, B766 (1965).
• J. E. Mandula, Symmetries of the Relativistic Hydrogen Atom, Phys. Rev. 185, 1774 (1969).
• W. D. McGlinn, Problem of Combining Interaction Symmetries and Relativistic Invariance, Phys. Rev. Lett. 12, 467 (1964).
• L. Michel, Relations Between Internal Symmetry and Relativistic Invariance, Phys. Rev. 137, B405 (1965).
• L. Michel and B. Sakita, Group of Invariance of a Relativistic Supermultiplet Theory, Ann. Inst. Henri Poincaré 2, 167 (1965).
• T. Nakano and K. Nishijima, Charge Independence for V-particles, Progress of Theoretical Physics 10, 581 (1953).
• Y. Ne’eman, Derivation of Strong Interactions from a Gauge Invariance, Nucl. Phys. 26, 222 (1961).
• S. Okubo, Note on Unitary Symmetry in Strong Interactions, Prog. Theo. Phys. 27 (5), 949 (1962).
• A. Pais, Implications of Spin-Unitary Spin Independence, Phys. Rev. Lett. 13, 175 (1964).
• B. Sakita, Supermultiplets of elementary particles, Phys. Rev. 136, B1756 (1964).
• S. Weinberg, Comments on Relativistic Supermultiplet Theories, Phys. Rev. 139, B597 (1965).
• S. Weinberg, "The Quantum Theory of Fields: Volume III: Supersymmetry". See esp. pp 1-24 (Cambridge University Press, New York, 2000).
• J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B, 70, 39 (1974a).
• J. Wess and B. Zumino, A Lagrangian Model Invariant Under Supergauge Transformations, Phys. Lett. 49B, 52 (1974b).
• E. Wigner, On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei, Phys. Rev. 51, 106 (1937).

• S. Coleman, Trouble with Relativistic SU(6), Phys. Rev. 138, B1262 (1965).
• S. Coleman and J. Mandula, All Possible Symmetries of the S Matrix, Phys. Rev. 159, 1251 (1967).
• R. Haag, J. Łopuszański, and M. Sohnius, All Possible Generators of Supersymmetries of the S Matrix, Nucl. Phys. B, 88, 257 (1975).
• S. Weinberg, "The Quantum Theory of Fields: Volume III: Supersymmetry". (Esp. Chaps. 24 and 25) (Cambridge University Press, New York, 2000).


text:
# Consider the transformation T : <mi mathvariant="double-struck">R 3 </msup

Consider the transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{2}$ defined by:
$T\left(x\right)=T\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(2{x}_{1}+{x}_{3}\right)\left(\begin{array}{c}1\\ 2\end{array}\right)+\left({x}_{2}-3{x}_{3}\right)\left(\begin{array}{c}-1\\ 1\end{array}\right)$
1a) determine the matrix of the above transformation
1b) determine the reduced row echelon form of the matrix found in 1a
1c) based on your answer to part 1b, is the transformation T onto?
1d) based on your answer to part 1b, is the transformation T one-to-one?
1e) based on your answer to part 1b, determine the set of vectors $x$ in ${\mathbb{R}}^{3}$ for which $T\left(x\right)=0$. Write your answer in parametric vector form
You can still ask an expert for help

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postojahob
For part (a), notice that
$T\left(x\right)=\left(2{x}_{1}+{x}_{3}\right)\left[\begin{array}{c}1\\ 2\end{array}\right]+\left({x}_{2}-3{x}_{3}\right)\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}2{x}_{1}+{x}_{3}\\ 4{x}_{1}+2{x}_{3}\end{array}\right]+\left[\begin{array}{c}-{x}_{2}+3{x}_{3}\\ {x}_{2}-3{x}_{3}\end{array}\right]=\left[\begin{array}{c}2{x}_{1}-{x}_{2}+4{x}_{3}\\ 4{x}_{1}+{x}_{2}-{x}_{3}\end{array}\right]=\left[\begin{array}{ccc}2& -1& 4\\ 4& 1& -1\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]$
Which implies that the matrix defining $T\left(x\right)$ is
$\left[\begin{array}{ccc}2& -1& 4\\ 4& 1& -1\end{array}\right]$
For 1e) I'm getting the reduced-form matrix as :
$\left[\begin{array}{ccc}2& -1& 4\\ 4& 1& -1\end{array}\right]\to \left[\begin{array}{ccc}1& 0& \frac{1}{2}\\ 0& 1& -3\end{array}\right]$
Which gives the system of equations
${x}_{1}+\frac{1}{2}{x}_{3}=0$
${x}_{2}-3{x}_{3}=0$
Which implies that
${x}_{1}=-\frac{1}{2}{x}_{3}$
${x}_{2}=3{x}_{3}$
Lastly, if we define ${x}_{3}=t$, to solution in parametric form is then
${x}_{1}=-\frac{1}{2}t$
${x}_{2}=3t$
${x}_{3}=t$
###### Not exactly what you’re looking for?
fythynwyrk0
For part a), it helps to rewrite
$T\left(x\right)=\left(\begin{array}{c}2{x}_{1}-{x}_{2}+4{x}_{3}\\ 4{x}_{1}+{x}_{2}-{x}_{3}\end{array}\right)$
Now, for which matrix $A$ do we have $T\left(x\right)=Ax$?
If you go through the rest of the problem correctly, you should find that the transformation is onto, but not one-to-one. Solving part e) is the same thing as finding the nullspace of the matrix.


text:
# optimPPL: Optimization of sample configurations for variogram... In spsann: Optimization of Sample Configurations using Spatial Simulated Annealing

## Description

Optimize a sample configuration for variogram identification and estimation. A criterion is defined so that the optimized sample configuration has a given number of points or point-pairs contributing to each lag-distance class (PPL).

## Usage

  1 2 3 4 5 6 7 8 9 10 11 optimPPL(points, candi, lags = 7, lags.type = "exponential", lags.base = 2, cutoff, criterion = "distribution", distri, pairs = FALSE, schedule = scheduleSPSANN(), plotit = FALSE, track = FALSE, boundary, progress = "txt", verbose = FALSE) objPPL(points, candi, lags = 7, lags.type = "exponential", lags.base = 2, cutoff, distri, criterion = "distribution", pairs = FALSE, x.max, x.min, y.max, y.min) countPPL(points, candi, lags = 7, lags.type = "exponential", lags.base = 2, cutoff, pairs = FALSE, x.max, x.min, y.max, y.min)

## Arguments

 points Integer value, integer vector, data frame or matrix, or list. Integer value. The number of points. These points will be randomly sampled from candi to form the starting sample configuration. Integer vector. The row indexes of candi that correspond to the points that form the starting sample configuration. The length of the vector defines the number of points. Data frame or matrix. An object with three columns in the following order: [, "id"], the row indexes of candi that correspond to each point, [, "x"], the projected x-coordinates, and [, "y"], the projected y-coordinates. List. An object with two named sub-arguments: fixed, a data frame or matrix with the projected x- and y-coordinates of the existing sample configuration – kept fixed during the optimization –, and free, an integer value defining the number of points that should be added to the existing sample configuration – free to move during the optimization. candi Data frame or matrix with the candidate locations for the jittered points. candi must have two columns in the following order: [, "x"], the projected x-coordinates, and [, "y"], the projected y-coordinates. lags Integer value, the number of lag-distance classes. Alternatively, a vector of numeric values with the lower and upper bounds of each lag-distance class, the lowest value being larger than zero (e.g. 0.0001). Defaults to lags = 7. lags.type Character value, the type of lag-distance classes, with options "equidistant" and "exponential". Defaults to lags.type = "exponential". lags.base Numeric value, base of the exponential expression used to create exponentially spaced lag-distance classes. Used only when lags.type = "exponential". Defaults to lags.base = 2. cutoff Numeric value, the maximum distance up to which lag-distance classes are created. Used only when lags is an integer value. If missing, it is set to be equal to the length of the diagonal of the rectangle with sides x.max and y.max as defined in scheduleSPSANN. criterion Character value, the feature used to describe the energy state of the system configuration, with options "minimum" and "distribution". Defaults to objective = "distribution". distri Numeric vector, the distribution of points or point-pairs per lag-distance class that should be attained at the end of the optimization. Used only when criterion = "distribution". Defaults to a uniform distribution. pairs Logical value. Should the sample configuration be optimized regarding the number of point-pairs per lag-distance class? Defaults to pairs = FALSE. schedule List with 11 named sub-arguments defining the control parameters of the cooling schedule. See scheduleSPSANN. plotit (Optional) Logical for plotting the optimization results, including a) the progress of the objective function, and b) the starting (gray circles) and current sample configuration (black dots), and the maximum jitter in the x- and y-coordinates. The plots are updated at each 10 jitters. When adding points to an existing sample configuration, fixed points are indicated using black crosses. Defaults to plotit = FALSE. track (Optional) Logical value. Should the evolution of the energy state be recorded and returned along with the result? If track = FALSE (the default), only the starting and ending energy states are returned along with the results. boundary (Optional) SpatialPolygon defining the boundary of the spatial domain. If missing and plotit = TRUE, boundary is estimated from candi. progress (Optional) Type of progress bar that should be used, with options "txt", for a text progress bar in the R console, "tk", to put up a Tk progress bar widget, and NULL to omit the progress bar. A Tk progress bar widget is useful when using parallel processors. Defaults to progress = "txt". verbose (Optional) Logical for printing messages about the progress of the optimization. Defaults to verbose = FALSE. x.max Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum distance between two neighbouring candidate locations. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi. x.min Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum distance between two neighbouring candidate locations. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi. y.max Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum distance between two neighbouring candidate locations. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi. y.min Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum distance between two neighbouring candidate locations. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi.

## Details

Details about the mechanism used to generate a new sample configuration out of the current sample configuration by randomly perturbing the coordinates of a sample point are available in the help page of spJitter.

#### Lag-distance classes

Two types of lag-distance classes can be created by default. The first are evenly spaced lags (lags.type = "equidistant"). They are created by simply dividing the distance interval from 0.0001 to cutoff by the required number of lags. The minimum value of 0.0001 guarantees that a point does not form a pair with itself. The second type of lags is defined by exponential spacings (lags.type = "exponential"). The spacings are defined by the base b of the exponential expression b^n, where n is the required number of lags. The base is defined using the argument lags.base. See vgmLags for other details.

Using the default uniform distribution means that the number of point-pairs per lag-distance class (pairs = TRUE) is equal to n \times (n - 1) / (2 \times lag), where n is the total number of points and lag is the number of lags. If pairs = FALSE, then it means that the number of points per lag is equal to the total number of points. This is the same as expecting that each point contributes to every lag. Distributions other than the available options can be easily implemented changing the arguments lags and distri.

There are two optimizing criteria implemented. The first is called using criterion = "distribution" and is used to minimize the sum of the absolute differences between a pre-specified distribution and the observed distribution of points or point-pairs per lag-distance class. The second criterion is called using criterion = "minimum". It corresponds to maximizing the minimum number of points or point-pairs observed over all lag-distance classes.

## Value

optimPPL returns an object of class OptimizedSampleConfiguration: the optimized sample configuration with details about the optimization.

objPPL returns a numeric value: the energy state of the sample configuration – the objective function value.

countPPL returns a data.frame with three columns: a) the lower and b) upper limits of each lag-distance class, and c) the number of points or point-pairs per lag-distance class.

## Note

The distance between two points is computed as the Euclidean distance between them. This computation assumes that the optimization is operating in the two-dimensional Euclidean space, i.e. the coordinates of the sample points and candidate locations should not be provided as latitude/longitude. spsann has no mechanism to check if the coordinates are projected: the user is responsible for making sure that this requirement is attained.

## Author(s)

Alessandro Samuel-Rosa alessandrosamuelrosa@gmail.com

## References

Bresler, E.; Green, R. E. Soil parameters and sampling scheme for characterizing soil hydraulic properties of a watershed. Honolulu: University of Hawaii at Manoa, p. 42, 1982.

Pettitt, A. N.; McBratney, A. B. Sampling designs for estimating spatial variance components. Applied Statistics. v. 42, p. 185, 1993.

Russo, D. Design of an optimal sampling network for estimating the variogram. Soil Science Society of America Journal. v. 48, p. 708-716, 1984.

Truong, P. N.; Heuvelink, G. B. M.; Gosling, J. P. Web-based tool for expert elicitation of the variogram. Computers and Geosciences. v. 51, p. 390-399, 2013.

Warrick, A. W.; Myers, D. E. Optimization of sampling locations for variogram calculations. Water Resources Research. v. 23, p. 496-500, 1987.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 ## Not run: # This example takes more than 5 seconds require(sp) data(meuse.grid) candi <- meuse.grid[, 1:2] schedule <- scheduleSPSANN(chains = 1, initial.temperature = 30, x.max = 1540, y.max = 2060, x.min = 0, y.min = 0, cellsize = 40) set.seed(2001) res <- optimPPL(points = 10, candi = candi, schedule = schedule) objSPSANN(res) - objPPL(points = res, candi = candi) countPPL(points = res, candi = candi) ## End(Not run)

spsann documentation built on May 2, 2019, 1:36 p.m.