scholarly journals Special methods of analytic completion in field theory

1960 ◽  
Vol 256 (1284) ◽  
pp. 39-52 ◽  
Keyword(s):  
Field Theory ◽  
Integral Representation ◽  
Simple Proof ◽  
Vertex Function ◽  
Distribution Theory ◽  
Complex Variables ◽  
Scattering Function ◽  
Envelope Of Holomorphy

By combining known theorems in the theory of functions of many complex variables and distribution theory a technique of analytic completion is developed; this provides a simple proof of the ‘edge of the wedge* theorem. An integral representation for the double commutator is derived, and it is used to simplify some of the work involved in computing the envelope of holomorphy for the vertex function. The Jacobi identities have not been incorporated, with the consequence that the threefold case cannot be completely solved by this method. The technique is applied to the fourfold (scattering) function, again without the Jacobi identities being included. By this technique analytic completion can be performed for some but not all the domains encountered in the fourfold problem.

A theorem of continuation for functions of several complex variables

1958 ◽  
Vol 54 (3) ◽  
pp. 377-382 ◽  
Author(s):  
J. G. Taylor
Keyword(s):  
Field Theory ◽  
Circular Domain ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Quantum Field ◽  
Domain Of Holomorphy ◽  
Envelope Of Holomorphy ◽  
Special Cases ◽  
Theory Of Fields ◽  
Holomorphic Continuation

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

Quantum statistical physics: Functional integration formalism

2021 ◽  
pp. 64-89
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Phase Space ◽  
Integral Representation ◽  
Density Matrix ◽  
Thermal Equilibrium ◽  
Degrees Of Freedom ◽  
Functional Integral ◽  
Complex Variables ◽  
Path Integral Representation ◽  
Canonical Formulation ◽  
Grand Canonical

The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.

On the "Edge of the Wedge" Theorem

1963 ◽  
Vol 15 ◽  
pp. 125-131 ◽  
Author(s):  
Felix E. Browder
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Complex Variables ◽  
Point Of View ◽  
Several Complex Variables ◽  
Quantum Field ◽  
Complete Proof ◽  
Strategic Role ◽  
Mathematical Justification ◽  
Axiomatic Quantum Field Theory

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.

Sign in / Sign up

Export Citation Format

Share Document