Talk:Grassmann integral

message: 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)
thread_title:  Literature 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Candlin 

message: 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:
# 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.
# 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.
# 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.
# 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.
# 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)
thread_title:  Khalatnikov 

message: 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)
thread_title:  Khalatnikov from my point of view 

message: 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)
thread_title:  Berezin integral, not Grassmann integral (and the reply to the proposal of merging) 

message: 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)
thread_title:  Berezin integral was invented by Berezin