Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userShift Theorem Involving the Exponential of a Sum of Non-Commuting Operators in Path Integrals
46
0
0.0
(
0
)
Authors
Fred Cooper
Ask ChatGPT about the research
No Arabic abstract
We consider expressions of the form of an exponential of the sum of two non-commuting operators of a single variable inside a path integration. We show that it is possible to shift one of the non-commuting operators from the exponential to other functions which are pre-factors and post-factors when the domain of integration of the argument of that function is from -infty to +infty. This shift theorem is useful to perform certain integrals and path integrals involving the exponential of sum of two non-commuting operators.
rate research
Read More
For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.
A salient feature of the Schr{o}dinger equation is that the classical radial momentum term $p_{r}^{2}$ in polar coordinates is replaced by the operator $hat{P}^{dagger}_{r} hat{P}_{r}$, where the operator $hat{P}_{r}$ is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator $hat{p}_{r}=(1/2)((frac{hat{vec{x}}}{r}) hat{vec{p}}+hat{vec{p}}(frac{hat{vec{x}}}{r}))$, the relation $hat{P}^{dagger}_{r} hat{P}_{r}=hat{p}_{r}^{2}+hbar^{2}(d-1)(d-3)/(4r^{2})$ holds in $d$-dimensional space and this extra potential appears in the path integral formulated in polar coordinates. The extra potential, which influences the classical solutions in the semi-classical treatment such as in the analysis of solitons and collective modes, vanishes for $d=3$ and attractive for $d=2$ and repulsive for all other cases $dgeq 4$. This extra term induced by the non-hermitian operator is a purely quantum effect, and it is somewhat analogous to the quantum anomaly in chiral gauge theory.
Perturbative quantum field theory usually uses second quantisation and Feynman diagrams. The worldline formalism provides an alternative approach based on first quantised particle path integrals, similar in spirit to string perturbation theory. Here we review the history, main features and present applications of the formalism. Our emphasis is on recent developments such as the path integral representation of open fermion lines, the description of colour using auxiliary worldline fields, incorporation of higher spin, and extension of the formalism to non-commutative space.
Time derivatives of scalar fields occur quadratically in textbook actions. A simple Legendre transformation turns the lagrangian into a hamiltonian that is quadratic in the momenta. The path integral over the momenta is gaussian. Mean values of operators are euclidian path integrals of their classical counterparts with positive weight functions. Monte Carlo simulations can estimate such mean values. This familiar framework falls apart when the time derivatives do not occur quadratically. The Legendre transformation becomes difficult or so intractable that one cant find the hamiltonian. Even if one finds the hamiltonian, it usually is so complicated that one cant path-integrate over the momenta and get a euclidian path integral with a positive weight function. Monte Carlo simulations dont work when the weight function assumes negative or complex values. This paper solves both problems. It shows how to make path integrals without knowing the hamiltonian. It also shows how to estimate complex path integrals by combining the Monte Carlo method with parallel numerical integration and a look-up table. This Atlantic City method lets one estimate the energy densities of theories that, unlike those with quadratic time derivatives, may have finite energy densities. It may lead to a theory of dark energy. The approximation of multiple integrals over weight functions that assume negative or complex values is the long-standing sign problem. The Atlantic City method solves it for problems in which numerical integration leads to a positive weight function.
A number of irreducible master integrals for L-loop sunrise-type and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via Mellin-Barnes representation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
