Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime

C. Gérard; M. Wrochna

doi:10.1007/s00220-015-2305-0

Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime

Reviews in Mathematical Physics ◽

10.1142/s0129055x17500143 ◽

2017 ◽

Vol 29 (04) ◽

pp. 1750014 ◽

Cited By ~ 4

Author(s):

Michał Wrochna ◽

Jochen Zahn

Keyword(s):

Phase Space ◽

Cauchy Surface ◽

Gauge Field Theories ◽

Subsidiary Condition ◽

Yang Mills ◽

Special Cases ◽

Hadamard States ◽

Hyperbolic Spacetimes ◽

Definition Of ◽

Classical Phase Space

We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the ‘subsidiary condition’ approach of Hack and Schenkel in the case of Maxwell, Yang–Mills, and Rarita–Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang–Mills case is concluded from known results in the subsidiary condition (or Gupta–Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang–Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface.

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DYNAMICAL BACKREACTION IN ROBERTSON–WALKER SPACETIME

Reviews in Mathematical Physics ◽

10.1142/s0129055x11004357 ◽

2011 ◽

Vol 23 (05) ◽

pp. 531-551 ◽

Cited By ~ 5

Author(s):

BENJAMIN ELTZNER ◽

HANNO GOTTSCHALK

Keyword(s):

Dynamical System ◽

Scalar Field ◽

Stress Tensor ◽

Einstein Equation ◽

Scale Parameter ◽

The State ◽

Curved Spacetime ◽

Free Scalar ◽

Hadamard States ◽

Quantized Field

The treatment of a quantized field in a curved spacetime requires the study of backreaction of the field on the spacetime via the semiclassical Einstein equation. We consider a free scalar field in spatially flat Robertson–Walker spacetime. We require the state of the field to allow for a renormalized semiclassical stress tensor. We calculate the singularities of the stress tensor restricted to equal times in agreement with the usual renormalization prescription for Hadamard states to perform an explicit renormalization. The dynamical system for the Robertson–Walker scale parameter a(t) coupled to the scalar field is finally derived for the case of conformal and also general coupling.

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INFLATION FROM AN EFFECTIVE STANDARD MODEL OF PARTICLE PHYSICS FOR CURVED SPACETIME

International Journal of Modern Physics D ◽

10.1142/s0218271899000250 ◽

1999 ◽

Vol 08 (03) ◽

pp. 337-347

Author(s):

TONATIUH MATOS ◽

GUILLERMO ARREAGA ◽

GABRIELLA PICCINELLI

Keyword(s):

Standard Model ◽

Particle Physics ◽

Electroweak Interaction ◽

Higgs Field ◽

Effective Theory ◽

Inflationary Cosmology ◽

Curved Spacetime ◽

Field Equations ◽

Yang Mills ◽

Qualitative Discussion

Beginning from an effective theory in eight dimensions, in Ref. 1, Macias, Camacho and Matos proposed an effective model for the electroweak part of the Standard Model of particles in curved spacetime. Using this model, we investigate the cosmological consequences of the electroweak interaction in the early universe. We use the approximation that, near the Planck epoch, the Yang–Mills fields behave like a perfect fluid. Then we recover the field equations of inflationary cosmology, with the Higgs field directly related to the inflaton. We present some qualitative discussion about this and analyze the behavior of isospin space using some known exact solutions.

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MICROLOCAL SPECTRUM CONDITION AND HADAMARD FORM FOR VECTOR-VALUED QUANTUM FIELDS IN CURVED SPACETIME

Reviews in Mathematical Physics ◽

10.1142/s0129055x01001010 ◽

2001 ◽

Vol 13 (10) ◽

pp. 1203-1246 ◽

Cited By ~ 61

Author(s):

HANNO SAHLMANN ◽

RAINER VERCH

Keyword(s):

Vector Bundle ◽

Scalar Fields ◽

Quantum Fields ◽

Curved Spacetime ◽

Globally Hyperbolic ◽

Commutation Relations ◽

Hyperbolic Spacetime ◽

Hadamard Condition ◽

Spectrum Condition ◽

Hadamard States

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed "wavefront set spectrum condition"), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance saling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.

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ERRATUM TO "HADAMARD STATES, ADIABATIC VACUA AND THE CONSTRUCTION OF PHYSICAL STATES FOR SCALAR QUANTUM FIELDS ON CURVED SPACETIME"

Reviews in Mathematical Physics ◽

10.1142/s0129055x02001326 ◽

2002 ◽

Vol 14 (05) ◽

pp. 511-517 ◽

Cited By ~ 9

Author(s):

WOLFGANG JUNKER

Keyword(s):

Quantum Fields ◽

Curved Spacetime ◽

Scalar Quantum ◽

Hadamard States ◽

Math Phys

In this erratum we want to correct or modify some of the original statements and proofs in "Hadamard States, Adiabatic Vacua and the Construction of Physical States for Scalar Quantum Fields on Curved Spacetime" in Rev. Math. Phys. 8 (1996) 1091–1159.

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