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

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.

Download Full-text

The art of the state

International Journal of Modern Physics D ◽

10.1142/s0218271818430071 ◽

2018 ◽

Vol 27 (11) ◽

pp. 1843007 ◽

Cited By ~ 4

Author(s):

Christopher J. Fewster

Keyword(s):

Infinite Family ◽

Vacuum State ◽

Natural Conditions ◽

Spacetime Geometry ◽

High Energies ◽

Hyperbolic Spacetime ◽

Hadamard Condition ◽

Free Scalar ◽

Model Independent ◽

Hadamard States

Quantum field theory (QFT) on curved spacetimes lacks an obvious distinguished vacuum state. We review a recent no-go theorem that establishes the impossibility of finding a preferred state in each globally hyperbolic spacetime, subject to certain natural conditions. The result applies in particular to the free scalar field, but the proof is model-independent and therefore of wider applicability. In addition, we critically examine the recently proposed “SJ states”, that are determined by the spacetime geometry alone, but which fail to be Hadamard in general. We describe a modified construction that can yield an infinite family of Hadamard states, and also explain recent results that motivate the Hadamard condition without direct reference to ultra-high energies or ultra-short distance structure.

Download Full-text

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

Reviews in Mathematical Physics ◽

10.1142/s0129055x97000233 ◽

1997 ◽

Vol 09 (05) ◽

pp. 635-674 ◽

Cited By ~ 20

Author(s):

Rainer Verch

Keyword(s):

Symplectic Form ◽

Scalar Product ◽

Classical System ◽

Symplectic Space ◽

Haag Duality ◽

Globally Hyperbolic ◽

Classical Energy ◽

Hyperbolic Spacetime ◽

Klein Gordon ◽

Hadamard States

We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

Download Full-text

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.

Download Full-text

EFFECTS OF A BOUNDARY ON SPONTANEOUS EXCITATION OF AN ACCELERATED MULTILEVEL ATOM COUPLED WITH THE DERIVATIVE OF SCALAR FIELDS

International Journal of Modern Physics A ◽

10.1142/s0217751x07036804 ◽

2007 ◽

Vol 22 (20) ◽

pp. 3447-3461

Author(s):

YUNFENG ZHU ◽

HONGWEI YU

Keyword(s):

Radiation Reaction ◽

Scalar Fields ◽

Normal Direction ◽

Coupling Scheme ◽

Quantum Fields ◽

Vacuum Fluctuations ◽

Direction Parallel ◽

Excitation Rate ◽

Derivative Coupling ◽

Multilevel Atom

The presence of boundaries modifies the modes of quantum fields, which may in turn modifies the spontaneous excitation rate of accelerated atoms in interaction with these fields. In this paper, we study the effect of the presence of a reflecting boundary on the spontaneous excitation of a uniformly accelerated polarized multilevel atom interacting with quantum scalar fields in a dipole-derivative coupling scheme. We separately calculate the contributions of modified vacuum fluctuations and the radiation reaction to the spontaneous excitation rate of the atom. Our results show that the presence of the boundary modulates the excitation rate and makes it a function of the atom's distance from the boundary. When the atom is placed closer and closer to the boundary, the influence of the boundary becomes more and more drastic, with the contribution of the atom's polarization in the direction parallel to the boundary to the spontaneous excitation rate dramatically suppressed while that in the normal direction greatly enhanced.

Download Full-text

Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. II. Irreducible Representations

Journal of Mathematical Physics ◽

10.1063/1.1665050 ◽

1970 ◽

Vol 11 (1) ◽

pp. 21-29 ◽

Cited By ~ 2

Author(s):

Gerhard C. Hegerfeldt

Keyword(s):

Irreducible Representations ◽

Quantum Fields ◽

Canonical Commutation Relations ◽

Commutation Relations ◽

Integral Formulas

Download Full-text

QUANTUM FIELDS WITH MODIFIED DISPERSION RELATIONS IN CURVED SPACES

International Journal of Modern Physics D ◽

10.1142/s0218271811019086 ◽

2011 ◽

Vol 20 (05) ◽

pp. 745-756 ◽

Cited By ~ 2

Author(s):

FRANCISCO DIEGO MAZZITELLI

Keyword(s):

Vector Field ◽

General Structure ◽

Scalar Fields ◽

Field Approximation ◽

Weak Field ◽

Dispersion Relations ◽

Quantum Fields ◽

Renormalization Procedure ◽

Modified Dispersion Relations ◽

Weak Field Approximation

We discuss the renormalization procedure for quantum scalar fields with modified dispersion relations in curved spacetimes. We consider two different ways of introducing modified dispersion relations: through the interaction with a dynamical temporal vector field, as in the context of the Einstein–Aether theory, and breaking explicitly the covariance of the theory, as in Hǒrava–Lifshitz gravity. Working in the weak field approximation, we show that the general structure of the counterterms depends on the UV behavior of the dispersion relations and on the mechanism chosen to introduce them.

Download Full-text

Classical fields in curved spacetime

Introduction to Quantum Field Theory with Applications to Quantum Gravity ◽

10.1093/oso/9780198838319.003.0012 ◽

2021 ◽

pp. 294-313

Author(s):

Iosif L. Buchbinder ◽

Ilya L. Shapiro

Keyword(s):

Vector Fields ◽

Lorentz Invariance ◽

Conformal Symmetry ◽

Scalar Fields ◽

Gauge Fields ◽

Curved Spacetime ◽

Curved Space ◽

Spinor Fields ◽

Vector Gauge ◽

Classical Fields

This chapter discusses classical fields in an arbitrary Riemann spacetime. General considerations are followed by the formulation of scalar fields with non-minimal coupling. Spontaneous symmetry breaking in curved space is shown to provide the induced gravity action with a cosmological constant. The construction of spinor fields in curved spacetime is based on the notions of group theory from Part I and on the local Lorentz invariance. Massless vector fields (massless vector gauge fields) are described and the interactions between scalar, fermion and gauge fields formulated. A detailed discussion of classical conformal transformations and conformal symmetry for both matter fields and vacuum action is also provided.

Download Full-text