One Particle States in Curved Spacetime
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
One Particle States in Curved Spacetime Farhang Loran* Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran *e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—The traditional particle interpretation of states of quantum field theory in Minkowski spacetime relies on the Minkowski spacetime symmetry, i.e., the Poincare group uplifted to the Hilbert space. We argue that Weinberg’s approach to particles and interactions in Minkowski spacetime utilizes the representations of the inhomogeneous Lorentz transformations in the Hilbert space dissociated from the spacetime symmetry. Thus it provides a particle interpretation of states in general nonstationary curved spacetimes straightaway. We give Lagrangians for scalars, spinors and vector fields supporting the argument and show that these classical fields contribute to the cosmological constant. DOI: 10.1134/S154747712005026X
1. INTRODUCTION The particle interpretation of states in quantum field theory in Minkowski spacetime developed in [1] relies on inhomogeneous Lorentz transformations realized in Hilbert space. In curved spacetimes, local homogeneous Lorentz transformations in local frames can be used to assign spin to quantum fields. The corresponding states have a particle interpretation in stationary spacetimes, although the very existence of a meaningful particle interpretation of states in a general nonstationary curved spacetime has been an undecided issue yet [2]. In this work we show that Weinberg’s approach to particle physics extends to general D dimensional nonstationary curved spacetimes } endowed with a metric g immediately, once we reconsider his expressions as statements about quantum field theory in a coordinate system x in which det g = 1. In this interpretation of Weinber’s formulation, time-ordering still means ordering with respect to x 0 though the vector field ∂ 0 is not timelike all over, and quantum ∂x fields located at x1 and x2 (anti)commute for ηab ( x1 − x2 )a ( x1 − x2 )b < 0 , although Minkowski metric η = diag(1, −1, , −1) is not necessarily the metric of spacetime everywhere. These fields do not describe ordinary matter fields as can be inferred from their stress tensor which resemble a perfect fluid with equation of state w = −1. So they might be considered altogether as a source for the cosmological constant. We review the scalar field theory and explicate the x -coordinate system in Section 2. In Section 3 we discuss the background independence of Weinberg’s par-
ticle interpretation of quantum fields and give Lagrangians for free spin 1 , 1 fields whose second 2 quantization in nonstationary curved spacetime is available similarly to the second quantization of ordinary free field theory in Minkowski spacetime. 2. SCALAR CREEPERS
d -dimensional scalar creepers in D -dimensional Minkowski spacetime (} D , η) are scalar fields whose propagating modes are localized on d ≤ D dimensiona
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