scholarly journals Path integral representation of the evolution operator for the Dirac equation

2006 ◽  
Author(s):  
Alexander S. Lukyanenko ◽  
Inna A. Lukyanenko
Keyword(s):  
Dirac Equation ◽  
Integral Representation ◽  
Path Integral ◽  
Evolution Operator ◽  
Path Integral Representation

scholarly journals PATH INTEGRAL AND PSEUDOCLASSICAL ACTION FOR SPINNING PARTICLE IN EXTERNAL ELECTROMAGNETIC AND TORSION FIELDS

2000 ◽  
Vol 15 (24) ◽  
pp. 3861-3876 ◽  
Author(s):  
BODO GEYER ◽  
DMITRY GITMAN ◽  
ILYA L. SHAPIRO
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Integral Representation ◽  
Path Integral ◽  
Classical Limit ◽  
Equations Of Motion ◽  
Nonrelativistic Limit ◽  
Spinning Particle ◽  
Path Integral Representation ◽  
The Given

Starting from the Dirac equation in external electromagnetic and torsion fields we derive a path integral representation for the corresponding propagator. An effective action, which appears in the representation, is interpreted as a pseudoclassical action for a spinning particle. It is just a generalization of Berezin–Marinov action to the background under consideration. Pseudoclassical equations of motion in the nonrelativistic limit reproduce exactly the classical limit of the Pauli quantum mechanics in the same case. Quantization of the action appears to be nontrivial due to an ordering problem, which needs to be solved to construct operators of first-class constraints, and to select the physical sector. Finally the quantization reproduces the Dirac equation in the given background and, thus, justifies the interpretation of the action.

Quantum evolution: From particles to non-relativistic fields

2021 ◽  
pp. 90-104
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Integral Representation ◽  
Path Integral ◽  
Particle Physics ◽  
Evolution Operator ◽  
Scattering Amplitude ◽  
Eikonal Approximation ◽  
Functional Integral ◽  
Quantum Evolution ◽  
Path Integral Representation ◽  
Functional Integral Representation

Chapter 4 has introduced the functional integral representation of the quantum statistical operators and thus, formally, evolution in imaginary or Euclidean time. By contrast, to calculate the evolution operator and the scattering S-matrix elements, quantities relevant to particle physics, it is necessary to make a continuation from imaginary to real time. However, the representation of the S-matrix follows from additional considerations. To illustrate the power of the formalism, we show how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Chapter 4 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas, the evolution operator is then given by a holomorphic field integral. A parallel formalism leads to an analogous representation for the evolution operator of a system of non-relativistic fermions.

ANOTHER DERIVATION OF A SUPERPARTICLE ACTION

1991 ◽  
Vol 06 (21) ◽  
pp. 1977-1982 ◽  
Author(s):  
E. S. FRADKIN ◽  
SH. M. SHVARTSMAN
Keyword(s):  
Green Function ◽  
Integral Representation ◽  
Path Integral ◽  
Path Integral Representation ◽  
Chiral Superfield

It is shown that the reparametrization invariant superparticle action can be determined by constructing the path-integral representation for the causal Green function of a chiral superfield interacting with an external Maxwell superfield.

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