Quantum Field Theory in Terms of Vacuum Expectation Values

Abstract A precise link is derived between scalar-graviton S-matrix elements and expectation values of operators in a worldline quantum field theory (WQFT), both used to describe classical scattering of black holes. The link is formally provided by a worldline path integral representation of the graviton-dressed scalar propagator, which may be inserted into a traditional definition of the S-matrix in terms of time-ordered correlators. To calculate expectation values in the WQFT a new set of Feynman rules is introduced which treats the gravitational field hμν(x) and position $$ {x}_i^{\mu}\left({\tau}_i\right) $$ x i μ τ i of each black hole on equal footing. Using these both the 3PM three-body gravitational radiation 〈hμv(k)〉 and 2PM two-body deflection $$ \Delta {p}_i^{\mu } $$ Δ p i μ from classical black hole scattering events are obtained. The latter can also be obtained from the eikonal phase of a 2 → 2 scalar S-matrix, which we show corresponds to the free energy of the WQFT.

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The energy-momentum operator in curved space-time

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1983.0116 ◽

1983 ◽

Vol 389 (1797) ◽

pp. 379-403 ◽

Cited By ~ 37

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Space Time ◽

Momentum Operator ◽

Quantum Field ◽

Expectation Values ◽

Curved Space ◽

Renormalization Theory

We argue that the only meaningful geometrical measure of the energy—momentum of states of matter described by a free quantum field theory in a general curved space—time is that provided by a normal ordered energy-momentum operator. We contrast the finite expectation values of this operator with the conventional renormalized expectation values and further argue that the use of renormalization theory is inappropriate in this context.

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Double refraction phenomena in quantum field theory

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100033211 ◽

1958 ◽

Vol 54 (1) ◽

pp. 81-88 ◽

Cited By ~ 32

Author(s):

M. J. Stephen

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Optical Activity ◽

Kerr Effect ◽

Stokes Parameters ◽

Quantum Field ◽

Expectation Values ◽

Scattering Problems ◽

Double Refraction ◽

Usual Quantum

ABSTRACTIt is shown that the phenomena of double refraction—optical activity, Faraday rotation, Kerr effect, etc.—may be treated as scattering problems in quantum field theory. An expression for the scattering cross-section of an atom or molecule for photons may be easily obtained and from this expression using the idea of Stokes operators, equating their expectation values to the classical Stokes parameters, the usual quantum mechanical expressions for optical activity, etc. may be obtained. This approach does not require the ideas of refractive index or polarization of the medium, and emphasizes what are the actual observables in these experiments.

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QUANTUM FIELD THEORY IN CURVED SPACE–TIME, THE OPERATOR PRODUCT EXPANSION, AND DARK ENERGY

International Journal of Modern Physics D ◽

10.1142/s021827180801414x ◽

2008 ◽

Vol 17 (13n14) ◽

pp. 2607-2615 ◽

Cited By ~ 1

Author(s):

STEFAN HOLLANDS ◽

ROBERT M. WALD

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Operator Product Expansion ◽

Vacuum Expectation ◽

Vacuum State ◽

Space Time ◽

Operator Product ◽

Energy Tensor ◽

Quantum Field ◽

Curved Space

To make sense of quantum field theory in an arbitrary (globally hyperbolic) curved space–time, the theory must be formulated in a local and covariant manner in terms of locally measureable field observables. Since a generic curved space–time does not possess symmetries or a unique notion of a vacuum state, the theory also must be formulated in a manner that does not require symmetries or a preferred notion of a "vacuum state" and "particles". We propose such a formulation of quantum field theory, wherein the operator product expansion (OPE) of the quantum fields is elevated to a fundamental status, and the quantum field theory is viewed as being defined by its OPE. Since the OPE coefficients may be better behaved than any quantities having to do with states, we suggest that it may be possible to perturbatively construct the OPE coefficients — and, thus, the quantum field theory. By contrast, ground/vacuum states — in space–times, such as Minkowski space–time, where they may be defined — cannot vary analytically with the parameters of the theory. We argue that this implies that composite fields may acquire nonvanishing vacuum state expectation values due to nonperturbative effects. We speculate that this could account for the existence of a nonvanishing vacuum expectation value of the stress-energy tensor of a quantum field occurring at a scale much smaller than the natural scales of the theory.

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Perturbative quantum field theory (QFT): Algebraic methods

Quantum Field Theory and Critical Phenomena ◽

10.1093/oso/9780198834625.003.0007 ◽

2021 ◽

pp. 125-159

Author(s):

Jean Zinn-Justin

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Correlation Functions ◽

Feynman Diagrams ◽

Quantum Field ◽

Expectation Values ◽

Generating Functional ◽

Euclidean Time ◽

Functional Measure ◽

Scalar Quantum

This chapter discusses systematically the algebraic properties of perturbation theory in the example of a local, relativistic scalar quantum field theory (QFT). Although only scalar fields are considered, many results can be easily generalized to relativistic fermions. The Euclidean formulation of QFT, based on the density matrix at thermal equilibrium, is studied, mainly in the simpler zero-temperature limit, where all d coordinates, Euclidean time and space, can be treated symmetrically. The discussion is based on field integrals, which define a functional measure. The corresponding expectation values of product of fields called correlation functions are analytic continuations to imaginary (Euclidean) time of the vacuum expectation values of time-ordered products of field operators. They have also an interpretation as correlation functions in some models of classical statistical physics, in continuum formulations or, at equal time, of finite temperature QFT. The field integral, corresponding to an action to which a term linear in the field coupled to an external source J has been added, defines a generating functional Z(J) of field correlation functions. The functional W(J) = ln Z(J) is the generating functional of connected correlation functions, to which contribute only connected Feynman diagrams. In a local field theory connected correlation functions, as a consequence of locality, have cluster properties. The Legendre transform Γ(φ) [N1]of W(J) is the generating functional of vertex functions. To vertex functions contribute only one-line irreducible Feynman diagrams, also called one-particle irreducible (1PI).

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A NOVEL VARIATIONAL APPROACH FOR QUANTUM FIELD THEORY: EXAMPLE OF STUDY OF THE GROUND STATE AND PHASE TRANSITION IN NONLINEAR SIGMA MODEL

International Journal of Modern Physics A ◽

10.1142/s0217751x05026819 ◽

2005 ◽

Vol 20 (15) ◽

pp. 3488-3494 ◽

Cited By ~ 2

Author(s):

YURIY MISHCHENKO ◽

CHUENG-RYONG JI

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Ground State ◽

Variational Approach ◽

Sigma Model ◽

Fock Space ◽

Nonlinear Sigma Model ◽

Quantum Field ◽

Expectation Values ◽

Optimization Under Constraints

We discuss a novel form of the variational approach in Quantum Field Theory in which the trial quantum configuration is represented directly in terms of relevant expectation values rather than, e.g., increasingly complicated structure from Fock space. The quantum algebra imposes constraints on such expectation values so that the variational problem is formulated here as an optimization under constraints. As an example of application of such approach we consider the study of ground state and critical properties in a variant of nonlinear sigma model.

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