Photon fields in a fluctuating spacetime

Abstract

We present a model of interacting quantum fields, formulated in a non-perturbative manner. One of the fields is treated semi-classically, the other is the photon field. The model has an interpretation of an electromagnetic field in a fluctuating spacetime. The model is equivalent with the quantization of electromagnetism proposed recently by Czachor. Interesting features are that standard photon theory is recovered as a limiting case, and that localized field operators for the electromagnetic field exist as unbounded operators in Hilbert space.

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arXiv:hep-th/0210188v1 20 Oct 2002
Photon fields in a fluctuating spacetime
Jan Naudts◦, Maciej Kuna†,
and Wojciech De Roeck◦
◦Departement Natuurkunde, Universiteit Antwerpen UIA,
Universiteitsplein 1, 2610 Antwerpen, Belgium
†Wydzia? l Fizyki Technicznej i Matematyki Stosowanej,
Politechnika Gda´ nska,
ul. Narutowicza 11/12, 80-952 Gda´ nsk, Poland
E-mail: jan.naudts@ua.ac.be, maciek@mifgate.mif.pg.gda.pl,
wojciech.deroeck@ua.ac.be
v5, October 2002
Abstract
We present a model of interacting quantum fields, formulated in a
non-perturbative manner. One of the fields is treated semi-classically,
the other is the photon field. The model has an interpretation of an
electromagnetic field in a fluctuating spacetime.
The model is equivalent with the quantization of electromagnetism
proposed recently by Czachor. Interesting features are that standard
photon theory is recovered as a limiting case, and that localized field
operators for the electromagnetic field exist as unbounded operators
in Hilbert space.
1 Introduction
1.1 Two views on quantum spacetime
At the heart of quantum mechanics are the dual concepts of states and ob-
servables. One way to describe quantum spacetime is by means of a non-
abelian algebra of functions of position, or by introduction of non-commuting
position-time observables. Proposals for the latter go back to the early days
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of quantum mechanics [1]. This kind of approach is also the basis for applica-
tion of non-commutative geometry [2]. The alternative, studied in the present
paper, assumes that quantum spacetime has a state which is described by
a vector Ω in some Hilbert space H. This assumption implies that empty
spacetime has some structure, made up by fluctuations imposed on top of
Minkowski space. This structure is often referred to as the spacetime foam.
The concept of a fluctuating spacetime goes back to the work of Wheeler [3].
Let us assume that a shift of frame of the observer by a vector q in
Minkowski space is implemented by a unitary operator U(q). The shift mod-
ifies the state vector Ω into the new vector U(−q)Ω. One immediately expects
non-trivial autocorrelations of the form
w(q) = ?Ω|ˆU(−q)Ω?.(1)
However, since we cannot measure properties of spacetime without putting
particles in it, w(q) is not directly observable. In the standard concept of
spacetime the vector Ω equals the vacuum vector Ω0. The latter is invariant
under translations. Hence, in that case w(q) is constant equal to 1. If on the
other hand a spacetime structure exists on the scale of Planck’s length, or
some other relevant length scale l, then w(q) is non-trivial. It equals 1 for
q = 0 and vanishes for displacements, large compared to l. In Fourier space,
these short-distance correlations lead to a cutoff at high momenta.
1.2 Discussion
As is well known, the presence of a high-momentum cutoff implies breaking
of Poincar´ e invariance of the field theory. In the present case any Poincar´ e
transformation Λ maps the state Ω of spacetime onto another state ΛΩ. This
means that Poincar´ e covariance is maintained. Poincar´ e invariance is restored
at distances large compared to l by the assumption w(q) ≃ 0. Experimental
evidence for breaking of Poincar´ e invariance has been analysed using standard
theory [4]. If such a symmetry breaking exists it should be extremely small,
because a very small symmetry violation can produce dramatic observable
effects.
The presence of a spacetime structure on short length scales has effects
similar to those of noncommuting position-time operators. Indeed, a high-
momentum cutoff implies difficulties for measurements at short length scales.
However, here, as in most quantum field theories, it is not meaningful to
define position operators Qµ. Instead, one can determine field strengths
at positions q in Minkowsk space (see below). Quantum expectations of
these field strengths are affected by spacetime fluctuations. But it is difficult
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to conclude from inspection of these measurable quantities that spacetime
should be noncommutative.
In early attempts to quantize gravity the metric tensor gµν is treated
as a potential, analoguous to the vector potential of eletromagnetism. The
gravitons resulting from second quantization are then treated in the same
way as other field particles. In particular, gravitons are expected to interact
with all other particles. In the present model the interplay between spacetime
fluctuations and field particles is more fundamental and cannot be described
in terms of interactions. Field particles like photons or electrons cannot exist
without spacetime fluctuations. The structure of the total system is far from
the usual tensor product description of independent subsystems, which are
brought into interaction via lagrangian or hamiltonian constraints.
Another line of reasoning predicts fluctuations in the speed of light due to
quantum uncertainties of matter fields. This has been worked out in detail
by Ford et al [5, 6]. Here, speed of light is constant equal to 1. However, we
still have to investigate the effect of an inhomogeneous state of spacetime on
the effective propagation of electromagnetic radiation.
1.3Czachor’s noncanonical photon theory
As noted by several authors, there is some inconsistency between a descrip-
tion of the electromagnetic field by means of state vectors in a bosonic Fock
space, with a trivial vacuum state, and the regression to wavefunctions that
are ground states of the harmonic oscillator. Czachor et al [7, 8, 9] have
proposed to solve this discord by modifying the link between harmonic oscil-
lators and photons. As a first step, they replace all harmonic oscillators by
a single oscillator with quantized frequency, the eigenvalues of which are the
photon frequencies. In the next step, several copies of this quantized oscil-
lator are introduced. Photons become collective excitations of a spacetime
filled with oscillators.
Features of Czachor’s theory are that creation and annihilation operators
satisfy generalized commutation relations, and that the vacuum state is not
any longer unique. The present paper starts from rather different assump-
tions to arrive at the same theory, and hence, the same physics. In particular,
we know from Czachor’s work that the standard theory of quantized free elec-
tromagnetic fields is a limiting case of the present theory. Another property
of this theory is that field operatorsˆA(q) for photons or electrons exist as
genuine operators in Fock space, and not as operator valued distributions,
as is the case in the conventional theory. The latter is important for locality
of the electron-photon interaction. In the present formalism, the interaction
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term
?
R4
dqˆJµ(q)ˆAµ(q),(2)
with Jµ(q) the electron current, exists as an operator in Hilbert space.
1.4Structure of the paper
In the next section we formulate our model guided by symmetry considera-
tions. In section 3 a state of this model is described by means of its correlation
functions. Some of its properties are discussed. In section 4 the link is made
with the work of Czachor. The appendices contain a definition of covariance
systems and some technical proofs.
2A model of quantum spacetime
2.1 Symmetry considerations
An important symmetry of quantum fields is additivity of the fields. This
symmetry is central to the Weyl approach. Let H,+ denote the additive
group of single-particle wavefunctions. Then Fock space is a projective rep-
resentation of this group. Indeed, the Weyl operatorsˆW(ψ) satisfy the Weyl
form of the canonical commutation relations
ˆ W(φ)ˆ W(ψ) =ˆ W(φ + ψ)ζ(φ,ψ)(3)
with cocycle ζ given by
ζ(φ,ψ) = exp(iIm?φ|ψ?),(4)
and
?φ|ψ? =
?
R3
dk
1
2|k|φ(k)ψ(k).(5)
The field operatorˆA(ψ) is now the generator ofˆ W(ψ)
ˆ W(ψ) = exp
?
iˆA(ψ)
?
. (6)
The creation and annihilation operators are expressed in terms of the field
operators in the usual way
ˆA±(ψ) =1
2
ˆA(ψ) ±i
2
ˆA(iψ). (7)
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In the theory, presented below, additivity of the field of spacetime fluctu-
ations is a broken symmetry, as a consequence of the presence of additional
particle fields. Hence, the Weyl approach is not very well suited to describe
fields of spacetime fluctuations. This problem refrains us from formulating a
theory system based on the product of two additive groups, the group of fields
of spacetime fluctuations, and the group of additional particle fields. Instead,
we are forced to describe spacetime fluctuations by means of an algebra of
observables A. Its choice will be discussed later on. For the description of
the electromagnetic field we can continue to use the Weyl operator approach
as discussed above.
We do not include the Poincar´ e group in the initial formulation of the
theory. But this implies that Poincar´ e covariance has to be checked later on
a per state basis.
2.2 Covariance systems
It is obvious to assume that the fluctuations of spacetime have to be second-
quantized, and that they form a scalar boson field. Alternatively, a vector
or tensor field can be postulated. But this would complicate the theory. For
the moment we prefer to keep the model as simple as possible.
We use the correlation function approach of [10]. It starts from a co-
variance system, also known as C∗-dynamical system. States of this system
are determined by correlation functions. By means of a generalized G.N.S.-
theorem they determine a Hilbert space representation. See Appendix A.
The approach is rather analoguous to that of Wightman functions [11]. It
has been tested in a number of cases [12, 13]. In particular, it has been used
to give a clean treatment of the theory of free photons [14].
As discussed earlier, the spacetime fluctuations will be described by an
algebra A of observables, the electromagnetic field by an additive group
G of classical fields, or equivalently (see [14]), of test functions. This two
components can be combined into a covariance system by letting G act on
A in a trivial manner. Hence, the covariance system of choice is of the form
(A,G,I).
An obvious choice of algebra A would be the algebra of bounded operators
in the Fock space of a scalar boson. However, for technical reasons we are
forced to take the abelian subalgebra of functions of momenta. A consequence
is that in the present model the spacetime fluctuations are still described in
a semi-classical way.
More precisely, the C∗-algebra A consists of functions f(n)(k1,k2,···kn)
which depend on the number of particles n, and, for a given n, on wavevectors
km, m = 1,2,···n in R3. Because the particles are indistinguishable, invari-
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