Abstrct
We study the homogeneous non-Gaussian integral \( J_{n|r} (S) = \int {e^{ - S(x_1 , \ldots ,x_n )} } d^n x \), where S(x1,…,xn) is a symmetric form of degree r in n variables. This integral is naturally invariant under SL(n) transformations and therefore depends only on the invariants of the form. For example, in the case of quadratic forms, it is equal to the (−1/2)th power of the determinant of the form. For higher-degree forms, the integral can be calculated in some cases using the so-called Ward identities, which are second-order linear differential equations. We describe the method for calculating the integral and present detailed calculations in the case where n = 2 and r = 5. It is interesting that the answer is a hypergeometric function of the invariants of the form.
Similar content being viewed by others
References
A. Morozov and Sh. Shakirov, JHEP, 0912, 002 (2009); arXiv:0903.2595v1 [math-ph] (2009).
D. Hilbert, Theory of Algebraic Invariants, Cambridge Univ. Press, Cambridge (1993).
N. D. Beklemishev, Moscow Univ. Mat. Bull., 37, No. 2, 54–62 (1982).
H. Derksen and G. Kemper, Computational Invariant Theory (Encycl. Math. Sci., Vol. 130), Springer, Berlin (2002).
B. Sturmfels, Algorithms in Invariant Theory, Springer, Wien (2008).
V. Dolotin and A. Morozov, Introduction to Non-Linear Algebra, World Scientific, Hackensack, N. J. (2007); arXiv:hep-th/0609022v4 (2006).
V. Dolotin, “QFT’s with action of degree 3 and higher and degeneracy of tensors,” arXiv:hep-th/9706001v1 (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 163, No. 3, pp. 495–504, June, 2010.
Rights and permissions
About this article
Cite this article
Shakirov, S.R. Nonperturbative approach to finite-dimensional non-Gaussian integrals. Theor Math Phys 163, 804–812 (2010). https://doi.org/10.1007/s11232-010-0064-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-010-0064-9