Nonrelativistic limit of quantum field theory in inertial and noninertial frames and the principle of equivalence

Journal Article · Sat Oct 15 04:00:00 EDT 2011 · Physical Review. D, Particles Fields

DOI:https://doi.org/10.1103/PHYSREVD.84.085018· OSTI ID:21607951

Padmanabhan, Hamsa; Padmanabhan, T ^[1]

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1. Department of Physics, University of Pune, Ganeshkhind, Pune 411 007 (India)

We discuss the nonrelativistic limit of quantum field theory in an inertial frame, in the Rindler frame and in the presence of a weak gravitational field, and attempt to highlight and clarify several subtleties. In particular, we study the following issues: (a) While the action for a relativistic free particle is invariant under the Lorentz transformation, the corresponding action for a nonrelativistic free particle is not invariant under the Galilean transformation, but picks up extra contributions at the end points. This leads to an extra phase in the nonrelativistic wave function under a Galilean transformation, which can be related to the rest energy of the particle even in the nonrelativistic limit. We show that this is closely related to the peculiar fact that the relativistic action for a free particle remains invariant even if we restrict ourselves to O(1/c{sup 2}) in implementing the Lorentz transformation. (b) We provide a brief critique of the principle of equivalence in the quantum mechanical context. In particular, we show how solutions to the generally covariant Klein-Gordon equation in a noninertial frame, which has a time-dependent acceleration, reduce to the nonrelativistic wave function in the presence of an appropriate (time-dependent) gravitational field in the c{yields}{infinity} limit, and relate this fact to the validity of the principle of equivalence in a quantum mechanical context. We also show that the extra phase acquired by the nonrelativistic wave function in an accelerated frame, actually arises from the gravitational time dilation and survives in the nonrelativistic limit. (c) While the solution of the Schroedinger equation can be given an interpretation as being the probability amplitude for a single particle, such an interpretation fails in quantum field theory. We show how, in spite of this, one can explicitly evaluate the path integral using the (nonquadratic) action for a relativistic particle (involving a square root) and obtain the Feynman propagator. Further, we describe how this propagator reduces to the standard path integral kernel in the nonrelativistic limit. (d) We show that the limiting procedures for the propagators mentioned above work correctly even in the presence of a weak gravitational field, or in the Rindler frame, and discuss the implications for the principle of equivalence.

• OSTI ID: 21607951
• Journal Information: Physical Review. D, Particles Fields, Journal Name: Physical Review. D, Particles Fields Journal Issue: 8 Vol. 84; ISSN PRVDAQ; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ACCELERATION
DIFFERENTIAL EQUATIONS
ENERGY RANGE
EQUATIONS
FIELD EQUATIONS
FIELD THEORIES
FUNCTIONS
GALILEI TRANSFORMATIONS
GRAVITATIONAL FIELDS
KLEIN-GORDON EQUATION
LORENTZ TRANSFORMATIONS
MATHEMATICAL SOLUTIONS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM FIELD THEORY
QUANTUM MECHANICS
RELATIVISTIC RANGE
SCHROEDINGER EQUATION
TIME DEPENDENCE
TRANSFORMATIONS
WAVE EQUATIONS
WAVE FUNCTIONS