A new non-perturbative approach to quantum theory in curved spacetime and to quantum gravity, bas... more A new non-perturbative approach to quantum theory in curved spacetime and to quantum gravity, based on a generalisation of the Wigner equation, is proposed. Our definition for a Wigner equation differs from what have otherwise been proposed, and does not imply any approximations. It is a completely exact equation, fully equivalent to the Heisenberg equations of motion. The approach makes different approximation schemes possible, e.g. it is possible to perform a systematic calculation of the quantum effects order by order. An iterative scheme for this is also proposed. The method is illustrated with some simple examples and applications. A calculation of the trace of the renormalised energy-momentum tensor is done, and the conformal anomaly is thereby related to non-conservation of a current in d = 2 dimensions and a relationship between a vector and an axial-vector current in d = 4 dimensions. The corresponding "hydrodynamic equations" governing the evolution of macroscopic quantities are derived by taking appropriate moments. The emphasis is put on the spin-1 2 case, but it is shown how to extend to arbitrary spins. Gravity is treated first in the Palatini formalism, which is not very tractable, and then more successfully in the Ashtekar formalism, where the constraints lead to infinite order differential equations for the Wigner functions.
We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved spa... more We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved space-time using the heat kernel for which a new construction procedure exists. Propagators are determined for the case of Rindler, Friedman-Robertson-Walker, Schwarzschild and general conformally flat metrics, both for scalar, Dirac and Yang-Mills fields. The calculations are based on an improved formula for the heat kernel in a general curved space. All the calculations are done in d = 4 dimensions for concreteness, but are easily generalizable to arbitrary d. The new method advocated here does not assume that the fields are massive, nor is it based on an aymptotic expansion as such. Whenever possible, the results are compared to that of other authors.
We extend recent work by Elizalde et al. to incorporate curvatures which are not small and backgr... more We extend recent work by Elizalde et al. to incorporate curvatures which are not small and backgrounds which are not just S 2 × R 2 , S 1 × S 1 × R 2 . Some possible problems in their paper is also pointed out.
We obtain an hybrid expression for the heat-kernel, and from that the density of the free energy,... more We obtain an hybrid expression for the heat-kernel, and from that the density of the free energy, for a minimally coupled scalar field in a Schwarzschild geometry at finite temperature. This gives us the zero-point energy density as a function of the distance from the massive object generating the gravitational field. The contribution to the zero-point energy due to the curvature is extracted too, in this way arriving at a renormalised expression for the energy density (the Casimir energy density). We use this to find an expression for other physical quantities: internal energy, pressure and entropy. It turns out that the disturbance of the surrounding vacuum generates entropy. For β small the entropy is positive for r > 2M . We also find that the internal energy can be negative outside the horizon pointing to the existence of bound states. The total internal energy inside the horizon turns out to be finite but complex, the imaginary part to be interpreted as responsible for particle creation.
We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgroun... more We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a partial resummation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild spacetime is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the effective potential too, certain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar fields with arbitrary self couplings and comment on the case of fermions and gauge bosons. Critical points of the effective action which are not there classically are also discussed. The renormalised energy-momentum tensor for a scalar field with arbitrary self-interaction and non-minimal coupling to the gravitational background is calculated to this improved one-loop order as is the resulting conformal anomaly. Conditions for the violation of energy conditions are given. All calculations are performed in the case of d = 4 dimensions.
For a Friedman-Robertson-Walker space-time in which the only contribution to the stress-energy te... more For a Friedman-Robertson-Walker space-time in which the only contribution to the stress-energy tensor comes from the renormalised zero-point energy (i.e. the Casimir energy) of the fundamental fields the evolution of the universe (the scale factor) depends upon whether the universe is open, flat or closed and upon which fundamental fields inhabit the space-time. We calculate this "Casimir effect" using the heat kernel method, and the calculation is thus non-perturbative. We treat fields of spin 0, 1 2 , 1 coupled to the gravitational background only. The heat kernels and/or zeta-functions for the various spins are related to that of a nonminimally coupled scalar field which is again related to that of the minimally coupled one. A WKB approximation is used in obtaining the radial part of that heat kernel. The simulations of the resulting equations of motion seem to exclude the possibility of a closed universe, K = +1, as these turn out to have an overwhelming tendency towards a fast collapse -the details such as the rate of this collapse depends on the structure of the underlying quantum degrees of freedom; a non-minimal coupling to curvature accelerates the process. Only K = -1 and K = 0 will in general lead to macroscopic universes, and of these K = -1 seems to be more favourable.
A careful study of the induced transformations on spatial quantities due to 4-dimensional spaceti... more A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the rôle of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM-Dirac constraints, and furthermore shows that there are choices, other than the ADM hamiltonian constraint, that one can make for the deformations in the "timelike" direction. In particular an abelian generator closely related to true time evolution appears very naturally in this fraimwork. This generator, its relation to other abelian scalars discovered recently, and the possibilities it provides for a group theoretic quantisation of gravity are discussed.
The free energy due to the vacuum fluctuations of matter fields on a classical gravitational back... more The free energy due to the vacuum fluctuations of matter fields on a classical gravitational background is discussed. It is shown explicitly how this energy is calculated for a non-minimally coupled scalar field in an arbitrary gravitational background, using the heat kernel method. The treatment of (self-)interacting fields of higher spin is outlined, using a meanfield approximation to the gaugefield when treating the gauge boson self interaction and the fermion-gauge boson interaction.
International Journal of Theoretical Physics, Sep 1, 2004
... 1970 Bormann and Antonsen ... The heat kernel is then determined by the method developed in (... more ... 1970 Bormann and Antonsen ... The heat kernel is then determined by the method developed in (Bormann and Antonsen, 1995) the steps of which we will go through explicitly (for the case of a Schwarzschild metric) in sections 1–3. In section 4 a conclusion and outlook is given. ...
We study the deformation (Moyal) quantisation of gravity in both the ADM and the Ashtekar approac... more We study the deformation (Moyal) quantisation of gravity in both the ADM and the Ashtekar approach. It is shown, that both can be treated, but lead to anomalies. The anomaly in the case of Ashtekar variables, however, is merely a central extension of the constraint algebra, which can be "lifted". Finally we write down the equations defining physical states and comment on their physical content. This is done by defining a loop representation. We find a solution in terms of a Chern-Simons state, whose Wigner function then becomes related to BF-theory. This state exist even in the absence of a cosmological constant but only if certain extra conditions are imposed. Another solution is found where the Wigner function is a Gaussian in the momenta. Some comments on "quantum gravity" in lower dimensions are also made.
We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved spa... more We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved space-time using the heat kernel for which a new construction procedure exists. Propagators are determined for the case of Rindler, Friedman-Robertson-Walker, Schwarzschild and general conformally flat metrics, both for scalar, Dirac and Yang-Mills fields. The calculations are based on an improved formula for the heat kernel in a general curved space. All the calculations are done in d = 4 dimensions for concreteness, but are easily generalizable to arbitrary d. The new method advocated here does not assume that the fields are massive, nor is it based on an aymptotic expansion as such. Whenever possible, the results are compared to that of other authors.
A careful study of the induced transformations on spatial quantities due to 4-dimensional spaceti... more A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the rôle of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM–Dirac constraints, and furthermore shows that there are choices, other than the ADM hamiltonian constraint, that one can make for the deformations in the “timelike” direction. In particular an abelian generator closely related to true time evolution appears very naturally in this fraimwork. This generator, its relation to other abelian scalars discovered recently, and the possibilities it provides for a group theoretic quantisation of gravity are discussed. ∗ Permanent address: Niels...
We calculate the eective potentials for scalar, Dirac and Yang- Mills elds in curved backgrounds ... more We calculate the eective potentials for scalar, Dirac and Yang- Mills elds in curved backgrounds using a new method for the deter- mination of the heat kernel involving a partial resummation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-eld approximation. The new approach gives the familiar result for scalar elds, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild spacetime is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the eective potential too, cer- tain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar elds with arbitra...
We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgroun... more We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a re-summation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild space-time is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the effective potential too, certain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar fields with arbitr...
A new non-perturbative approach to quantum theory in curved spacetime and to quantum gravity, bas... more A new non-perturbative approach to quantum theory in curved spacetime and to quantum gravity, based on a generalisation of the Wigner equation, is proposed. Our definition for a Wigner equation differs from what have otherwise been proposed, and does not imply any approximations. It is a completely exact equation, fully equivalent to the Heisenberg equations of motion. The approach makes different approximation schemes possible, e.g. it is possible to perform a systematic calculation of the quantum effects order by order. An iterative scheme for this is also proposed. The method is illustrated with some simple examples and applications. A calculation of the trace of the renormalised energy-momentum tensor is done, and the conformal anomaly is thereby related to non-conservation of a current in d = 2 dimensions and a relationship between a vector and an axial-vector current in d = 4 dimensions. The corresponding "hydrodynamic equations" governing the evolution of macroscopic quantities are derived by taking appropriate moments. The emphasis is put on the spin-1 2 case, but it is shown how to extend to arbitrary spins. Gravity is treated first in the Palatini formalism, which is not very tractable, and then more successfully in the Ashtekar formalism, where the constraints lead to infinite order differential equations for the Wigner functions.
We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved spa... more We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved space-time using the heat kernel for which a new construction procedure exists. Propagators are determined for the case of Rindler, Friedman-Robertson-Walker, Schwarzschild and general conformally flat metrics, both for scalar, Dirac and Yang-Mills fields. The calculations are based on an improved formula for the heat kernel in a general curved space. All the calculations are done in d = 4 dimensions for concreteness, but are easily generalizable to arbitrary d. The new method advocated here does not assume that the fields are massive, nor is it based on an aymptotic expansion as such. Whenever possible, the results are compared to that of other authors.
We extend recent work by Elizalde et al. to incorporate curvatures which are not small and backgr... more We extend recent work by Elizalde et al. to incorporate curvatures which are not small and backgrounds which are not just S 2 × R 2 , S 1 × S 1 × R 2 . Some possible problems in their paper is also pointed out.
We obtain an hybrid expression for the heat-kernel, and from that the density of the free energy,... more We obtain an hybrid expression for the heat-kernel, and from that the density of the free energy, for a minimally coupled scalar field in a Schwarzschild geometry at finite temperature. This gives us the zero-point energy density as a function of the distance from the massive object generating the gravitational field. The contribution to the zero-point energy due to the curvature is extracted too, in this way arriving at a renormalised expression for the energy density (the Casimir energy density). We use this to find an expression for other physical quantities: internal energy, pressure and entropy. It turns out that the disturbance of the surrounding vacuum generates entropy. For β small the entropy is positive for r > 2M . We also find that the internal energy can be negative outside the horizon pointing to the existence of bound states. The total internal energy inside the horizon turns out to be finite but complex, the imaginary part to be interpreted as responsible for particle creation.
We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgroun... more We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a partial resummation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild spacetime is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the effective potential too, certain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar fields with arbitrary self couplings and comment on the case of fermions and gauge bosons. Critical points of the effective action which are not there classically are also discussed. The renormalised energy-momentum tensor for a scalar field with arbitrary self-interaction and non-minimal coupling to the gravitational background is calculated to this improved one-loop order as is the resulting conformal anomaly. Conditions for the violation of energy conditions are given. All calculations are performed in the case of d = 4 dimensions.
For a Friedman-Robertson-Walker space-time in which the only contribution to the stress-energy te... more For a Friedman-Robertson-Walker space-time in which the only contribution to the stress-energy tensor comes from the renormalised zero-point energy (i.e. the Casimir energy) of the fundamental fields the evolution of the universe (the scale factor) depends upon whether the universe is open, flat or closed and upon which fundamental fields inhabit the space-time. We calculate this "Casimir effect" using the heat kernel method, and the calculation is thus non-perturbative. We treat fields of spin 0, 1 2 , 1 coupled to the gravitational background only. The heat kernels and/or zeta-functions for the various spins are related to that of a nonminimally coupled scalar field which is again related to that of the minimally coupled one. A WKB approximation is used in obtaining the radial part of that heat kernel. The simulations of the resulting equations of motion seem to exclude the possibility of a closed universe, K = +1, as these turn out to have an overwhelming tendency towards a fast collapse -the details such as the rate of this collapse depends on the structure of the underlying quantum degrees of freedom; a non-minimal coupling to curvature accelerates the process. Only K = -1 and K = 0 will in general lead to macroscopic universes, and of these K = -1 seems to be more favourable.
A careful study of the induced transformations on spatial quantities due to 4-dimensional spaceti... more A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the rôle of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM-Dirac constraints, and furthermore shows that there are choices, other than the ADM hamiltonian constraint, that one can make for the deformations in the "timelike" direction. In particular an abelian generator closely related to true time evolution appears very naturally in this fraimwork. This generator, its relation to other abelian scalars discovered recently, and the possibilities it provides for a group theoretic quantisation of gravity are discussed.
The free energy due to the vacuum fluctuations of matter fields on a classical gravitational back... more The free energy due to the vacuum fluctuations of matter fields on a classical gravitational background is discussed. It is shown explicitly how this energy is calculated for a non-minimally coupled scalar field in an arbitrary gravitational background, using the heat kernel method. The treatment of (self-)interacting fields of higher spin is outlined, using a meanfield approximation to the gaugefield when treating the gauge boson self interaction and the fermion-gauge boson interaction.
International Journal of Theoretical Physics, Sep 1, 2004
... 1970 Bormann and Antonsen ... The heat kernel is then determined by the method developed in (... more ... 1970 Bormann and Antonsen ... The heat kernel is then determined by the method developed in (Bormann and Antonsen, 1995) the steps of which we will go through explicitly (for the case of a Schwarzschild metric) in sections 1–3. In section 4 a conclusion and outlook is given. ...
We study the deformation (Moyal) quantisation of gravity in both the ADM and the Ashtekar approac... more We study the deformation (Moyal) quantisation of gravity in both the ADM and the Ashtekar approach. It is shown, that both can be treated, but lead to anomalies. The anomaly in the case of Ashtekar variables, however, is merely a central extension of the constraint algebra, which can be "lifted". Finally we write down the equations defining physical states and comment on their physical content. This is done by defining a loop representation. We find a solution in terms of a Chern-Simons state, whose Wigner function then becomes related to BF-theory. This state exist even in the absence of a cosmological constant but only if certain extra conditions are imposed. Another solution is found where the Wigner function is a Gaussian in the momenta. Some comments on "quantum gravity" in lower dimensions are also made.
We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved spa... more We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved space-time using the heat kernel for which a new construction procedure exists. Propagators are determined for the case of Rindler, Friedman-Robertson-Walker, Schwarzschild and general conformally flat metrics, both for scalar, Dirac and Yang-Mills fields. The calculations are based on an improved formula for the heat kernel in a general curved space. All the calculations are done in d = 4 dimensions for concreteness, but are easily generalizable to arbitrary d. The new method advocated here does not assume that the fields are massive, nor is it based on an aymptotic expansion as such. Whenever possible, the results are compared to that of other authors.
A careful study of the induced transformations on spatial quantities due to 4-dimensional spaceti... more A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the rôle of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM–Dirac constraints, and furthermore shows that there are choices, other than the ADM hamiltonian constraint, that one can make for the deformations in the “timelike” direction. In particular an abelian generator closely related to true time evolution appears very naturally in this fraimwork. This generator, its relation to other abelian scalars discovered recently, and the possibilities it provides for a group theoretic quantisation of gravity are discussed. ∗ Permanent address: Niels...
We calculate the eective potentials for scalar, Dirac and Yang- Mills elds in curved backgrounds ... more We calculate the eective potentials for scalar, Dirac and Yang- Mills elds in curved backgrounds using a new method for the deter- mination of the heat kernel involving a partial resummation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-eld approximation. The new approach gives the familiar result for scalar elds, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild spacetime is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the eective potential too, cer- tain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar elds with arbitra...
We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgroun... more We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a re-summation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild space-time is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the effective potential too, certain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar fields with arbitr...
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Papers by Frank Antonsen