Talk:Grassmann integral

Literature
Hi! I've added some interesting lines about Berezin integration in R^{0|n}. However I'm a bit weak in findind literature so I only gave one (really expensive) book. If anyone has another, possibly online, resource I would be thankful. [melli0620] 64.178.103.72 01:50, 15 August 2007 (UTC)

Candlin
Candlin invented this, but I only found a few obscure references. But his paper develops the whole theory. I don't know if he is still alive or any biographical details, but this is a central tool in modern physics--- it's 50% of the path integral, and the mathematically thornier half to boot. I gave him credit here, but he really should have his own page.Likebox (talk) 20:51, 27 August 2008 (UTC)


 * Also, most people call it a "Grassman integral", I suppose because Berezin didn't claim to invent it (although he did add tools). Perhaps the article should be retitled? Or merged with Grassman variable?Likebox (talk) 15:36, 28 August 2008 (UTC)


 * Most matematicians certainly call it "Berezin integral". As for Berezin, he of course "added tools", but before that he invented this integral. Cheers, Plf57 (talk) 13:52, 12 February 2011 (UTC)

= Berezin =

Hi. The two opinions of my esteemed colleagues: 1)`Candlin invented this' and 2)`most people call it a "Grassman integral"' seem a bit arbitrary and questionable. Our thorough search in literature shows that Candlin - being a very ingenious scientist of course - just introduced some artificial sums which, in some simple cases, could simulate something like Berezin integral, but not Berezin integral itself. In general, the contributions of Berezin predecessors - save Grassmann himself - can hardly be compared to the powerful theory of supermanifolds developed by Berezin.

Also, I think that mathematicians should be involved in this discussion, not only physicists. This can be useful in solving the question of "most people". The whole subject clearly belongs to both sciences.

Paloff (talk) 08:31, 14 October 2009 (UTC)


 * I agree that it belongs to both fields, however Candlin did not produce "artificial sums". He defines the notion of integration over a Grassman variable directly in his paper, and shows that the integral reproduces the sum over intermediate states necessary for a quantum mechanical path integral. This is a more complete treatment than even that found in Berezin's textbook.


 * Mathematicians have adopted the technique, but it is not found in Grassman's work or in any other 19th century work. The main idea, the integral, is not found in any work prior to Candlin '56.Likebox (talk) 03:32, 18 October 2009 (UTC)


 * OK, dear colleague Likebox, I must, according to Wikipedia's rules, assume that you are acting led by good intentions. However, of course there is *no* Berezin integral in Candlin's paper - have you, by the way, read it? Nor do all the further formulas that you are trying - from some very good intentions - ascribe to Candlin, belong to him.


 * This is why I say that a wide and *open* discussion is needed - will you be able to present your opinion about Candlin openly? The whole subject is by no means of "mid-importance", it is sometimes compared to introducing complex numbers in mathematics. My statement is: the two key figures are Grassmann and Berezin; others are less important.


 * So, please do dispute, but 1) be honest; 2) let you statements be not groundles.


 * With the very best wishes, Paloff (talk) 06:16, 18 October 2009 (UTC)


 * Of course I read the paper, and I do disagree with you: the main formulas of Grassman integration are found in Candlin's paper, including the translation invariance, the Fermionic coherent state justification, and the demonstration that the Fermionic path integral is equivalent to a sum over Fermionic states. I agree with the statement that this subject is comparable in importance to the introduction of Complex numbers. In that regard, Candlin defined that "i=-1" and showed how to do analytic continuation, while Berezin was like Euler, giving many extensions of the method. It is very important to understand that the entire theory is contained in Candlin '56.Likebox (talk) 16:09, 18 October 2009 (UTC)


 * "the entire theory is contained in Candlin '56" - can I ask you just to read, here at the page Berezin integral, what the theory of Berezin integral is about? Plf57 (talk) 18:14, 12 February 2011 (UTC)

Khalatnikov
JETP isn't scanned yet, so Khalatnikov's contribution, which seem to be closer to Candlin than Salam and Matthews (Feynman also does something similar to Salam and Matthews in one of the electrodynamics papers, but without the Fermionic coherent states, and without the justification in terms of sum over intermediate states, which are both found in Candlin for the first time, I think.) This paper talks about the result: the integral over Fermionic fields is defined following Schwinger so that the action principle will work, and the result is that the determinant is in the numerator and not the denominator. This is most of Grassman integration, but without the formal anticommuting variables, which seem to have been introduced by Berezin and Candlin independently (at least according to this source). I did not read the original Berezin articles, only the 1965 textbook, so perhaps they are more complete than I originally thought. If anyone has read these, please include a summary of the contents. The real question, in my mind, is as follows:


 * 1) Feynman's propagator approach only requires that the statistics be correct when permuting the ends of the paths. The rest can be done by summing over particle paths, with appropriate projections to deal with Dirac eqn. So the early authors, between 1948-1954 are all dealing with formal anticommuting variables (no complete theory). This is true of the Schwinger action principle, and to a lesser extent, of Matthews and Salam.
 * 2) Matthews and Salam, like Feynman in his QEDIII paper, treat the sum over histories as green's function manipulations, without trying to formally define a calculus of anticommuting fields. They do define anticommuting sources, and anticommuting field Green's functions, but this is similar to Feynman and Schwinger. It does not have the crucial addition that the algebra of the fields in space-time makes sense, and the Fermi field integral is a sum over intermediate states.
 * 3) Khalatnikov seems to have understood the modern thing: that the Fermionic integral is a concept that is well defined. Unfortunately, I cannot access his paper. If anyone has access, please find out what is in there.
 * 4) Candlin defines the whole thing, in a very modern way. This paper is the best on the subject I have read, but it is similar to Berezin 1964.
 * 5) Berezin has some papers in the 1950s and 1960s, which seem to develop the whole thing indepedently, starting with Khalatnikov's observation. These papers are also crucial--- who develops the Fermionic coherent states? And who shows that the Fermionic integral reproduces the sum over intermediate Fermionic states?

The whole thing is somewhat implicit in Matthews and Salam, but also in Schwinger and Feynman. The real important development is the identification of sum-over-intermediate states with grassmann integral. If someone can point out who did what, it would help a lot.Likebox (talk) 23:32, 20 October 2009 (UTC)

Khalatnikov from my point of view
As I don't have very much time at this moment, let me just say that as I *have* read Khalatnikov's paper attentively. It contains - I am saying this with full responsibility - a big algebraic blunder, the self-contradictory algebra of operators "eta". See text after formula (11), and formulas (12) and (13) in Khalatnikov's work - such operators obviously cannot exist. And this is important in the subsequent derivation of the formula (19) with the determinant in the numerator - but again, in fact, only part of this determinant is in the numerator, look at it attentively. Happily to Khalatnikov, the wrong factor cancels out when he divides by the vacuum expectation value.

Looks like these "eta"s have dynamic, so to say, properties, changing during the process of deriving the (already known from other sources) formula!

It also looks like Khalatnikov was too influential at the time (being an important figure in Landau's theoretical physics school), so neither Berezin nor Marinov could write openly about these glaring contradiction in his paper.

More philosophically, this story teaches us not to believe recklessly every person who says that he/she is a theoretical physicist. Paloff (talk) 06:48, 22 October 2009 (UTC)

Detailed analysis
Here is what I think. We must not make just unsubstantiated allegations, but present a detailed analysis of what all the authors we are mentioning have done. If my esteemed opponent Likebox thinks that Candlin did this and that, please kindly indicate the number of formula, the location of the relevant text fragment in his paper, etc. - something like I have done above for the paper of Khalatnikov. When I am typing this, I see right below the words "Encyclopedic content must be verifiable", mustn't it?

Or, when I have more time, I will do it myself. Candlin is surely a very ingenious scientist; the point is, in my opinion, that my opponent underestimates the importance of a mathematical theory. There are some very good insights in Candlin's paper; there is a mathematical theory in Berezin's works, and all the modern supersymmetry in theoretical physics is due to Berezin.

Best wishes, Paloff (talk) 15:34, 23 October 2009 (UTC)


 * I agree that Berezin should be credited with some aspects of modern supersymmetry methods, and that these are totally absent in Candlin. But the issue here is pure Berezin integration, just defining the integral over the anticommuting variables consistently. I wish I could get a hold of a copy of Khalatnikov's paper, to verify the mistake you point out.


 * I am not really an "opponent" here, just a second reader of the literature with a slightly different point of view. I just don't understand why Candlin's paper is not more widely known than it is. Berezin's methods are more developed, but Candlin has the main ideas. I think if there is a reasonable discussion, we will agree on how to present the history.


 * I haven't looked at Candlin's paper in a little while. I will reread it and post the key equations in a little bit.Likebox (talk) 22:37, 23 October 2009 (UTC)

Berezin integral, not Grassmann integral (and the reply to the proposal of merging)
Dear Colleagues,

As far as I know, the Berezin integral was invented by Berezin. So, the name "Grassmann integral" is incorrect, even though some people use it.

Of two current articles, "Berezin integral" is obviously much better than "Grassmann integral".

So, it makes sense just to delete "Grassmann integral", doesn't it?

Remark: of course, Grassmann was a great scientist, and if he, in reality, did invent this integral, I will take my words back. Also, Candlin was a great scientist and a forerunner of Berezin, but not inventor of the integral, strictly speaking.

I will be grateful for your opinions. Cheers, Plf57 (talk) 19:14, 8 February 2011 (UTC)

Berezin integral was invented by Berezin
I am undoing the change by a user who likes Candlin so much that ascribes to him the invention of the theory of Berezin integral. I have read Candlin's paper carefully: he only made some insights, this was not a mathematical theory. Candlin is a great scientist, though - but let us be fair and non-biased! Can I ask my colleagues to read the recent paper Felix Alexandrovich Berezin and his work before doing further improper changes in this article? Plf57 (talk) 16:30, 3 November 2012 (UTC)