Example of a form linear in infinitely many variables ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:11:13Z http://mathoverflow.net/feeds/question/105932 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105932/example-of-a-form-linear-in-infinitely-many-variables Example of a form linear in infinitely many variables ? Matthieu Romagny 2012-08-30T12:01:35Z 2012-09-02T21:37:04Z <p>We all know plenty of examples of multilinear forms in finitely many variables (e.g. determinants). However, I am missing an interesting example of a form in infinitely many variables, linear in each. The important point here is the word "interesting" ; the example should occur naturally in some nice situation in algebra, or geometry, or analysis... (One can easily find non-interesting examples.) I thought that maybe regularized determinants could provide examples, but I don't know much about these. Thanks for ideas !</p> http://mathoverflow.net/questions/105932/example-of-a-form-linear-in-infinitely-many-variables/105941#105941 Answer by Delio Mugnolo for Example of a form linear in infinitely many variables ? Delio Mugnolo 2012-08-30T12:47:37Z 2012-08-30T12:47:37Z <p>Well, if I understand you properly, what you are looking for is simply a linear mapping from an infinite dimensional vector space to R. What about $f\mapsto \int f(x) dx$?</p> http://mathoverflow.net/questions/105932/example-of-a-form-linear-in-infinitely-many-variables/105954#105954 Answer by Pavel Safronov for Example of a form linear in infinitely many variables ? Pavel Safronov 2012-08-30T15:09:08Z 2012-08-30T15:09:08Z <p>Consider the Hilbert space $H=L^2(S^1, \mathbf{C}^n)$ and the subspaces $H_+, H_-$ of positive and negative Fourier modes. One can construct the Hilbert space of infinite wedge products $V=\bigwedge(H_+)\hat{\otimes}\bigwedge(H_-)^*$.</p> <p>Just as $H$ carries an action of $LSU(n)=Maps(S^1, SU(n))$ by pointwise multiplication, $V$ carries an action of the central extension $\widetilde{LSU(n)}$. It turns out that all irreducible positive-energy level 1 representations occur as summands in $V$.</p> http://mathoverflow.net/questions/105932/example-of-a-form-linear-in-infinitely-many-variables/105969#105969 Answer by AlexArvanitakis for Example of a form linear in infinitely many variables ? AlexArvanitakis 2012-08-30T16:44:58Z 2012-09-02T21:37:04Z <p>Hmmm... I think that the functional determinants (as in <a href="http://en.wikipedia.org/wiki/Functional_determinant" rel="nofollow">http://en.wikipedia.org/wiki/Functional_determinant</a>) of Quantum Mechanics and Quantum Field Theory appear rather naturally. These CAN be defined rigorously, insofar as the defining Feynman path integrals are, in this case, defined rigorously. The most readable introduction to rigorous path integration I've read is "A Modern Approach to Functional Integration" by John R. Klauder.</p> <p>EDIT: I did some reading. The bosonic path integral expression (as in the wikipedia page I linked to earlier) for the functional determinant may fail to be multilinear (though it is rigorous), whereas the fermionic path integral expression</p> <p>$\det S = \int\int \exp (\langle \bar{c} | S | c\rangle) \mathcal D c \mathcal D \bar{c}$</p> <p>should be genuinely multilinear and valid for every "reasonable" S (including nonsymmetric/nonhermitian ones) IF fermionic functional integrals in infinite variables can be consistently defined to yield sensible results (e.g $\int 1 \mathcal D c=0$), which, as far as I know, has not yet been done rigorously.</p>