Eliminating unphysical photon components from DiracMaxwell Hamiltonian quantized in the Lorenz gauge
Abstract
We study the DiracMaxwell model quantized in the Lorenz gauge. In this gauge, the space of quantum mechanical state vectors inevitably be an indefinite metric vector space so that the canonical commutation relation (CCR) is realized in a Lorentz covariant manner. In order to obtain a physical subspace in which no negative norm state exists, the method first proposed by Gupta and Bleuler is applied with mathematical rigor. It is proved that a suitably defined physical subspace has a positive semidefinit metric, and naturally induces a physical Hilbert space with a positive definite metric. The original DiracMaxwell Hamiltonian naturally defines an induced Hamiltonian on the physical Hilbert space which is essentially selfadjoint.
1 Introduction
We consider a quantum system of Dirac particles under an external potential interacting with a quantized gauge field (so called DiracMaxwell model). If we apply the informal perturbation theory to this model, quantitative predictions are obtained such as the KleinNishina formula for the cross section of the Compton scattering of an electron and a photon [1], which agrees with the experimental results very well. Hence, the DiracMaxwell model is expected to describe a certain class of realistic quantum phenomena and thus is worth the analysis with mathematical rigor, even though it may suffer from so called “negative energy problem”. The mathematically rigorous study of this model was initiated by Arai in Ref. [2], and several mathematical aspects of the model was analyzed so far (see, e.g., Refs. [3], [4], [5], [6], and [7]).
The motivation of the present study is to treat this model in the Lorentz gauge in which the Lorenz covariance is manifest. In analyzing gauge theories such as quantum electrodynamics (QED) in a Lorentz covariant gauge, a difficulty always arises, since one inevitably adopt “an indefinite metric Hilbert space” as a space of all the state vectors (for instance, see Ref. [8]). In such cases, we have to identify a positive definite subspace as a physical state vector space by eliminating unphysical photon modes with negative norms. The most general and elegant method to identify the physical subspace for nonabelian gauge theories quantized in a covariant gauge was given by the celebrated work by Kugo and Ojima [9, 10], which is based on the BRST symmetry, the remnant gauge symmetry of the Lagrangian density after imposing some gauge fixing condition. The KugoOjima formulation reduces to Nakanishi and Lautrup’s field theory [11, 12, 13, 14] in the case where the gauge field is abelinan and after integrating out the auxiliary NakanishiLautrup’s field, it is reduced to the condition first proposed by Gupta and Bleuler [15, 16].
The Gupta and Bleuler condition says that a state vactor belongs to the phyisical subspace if and only if it has the vanishing expectation value of the operator , the four component divergence of the gauge field:
(1.1) 
at every spacetime point . From the equations of motion (with “mostly plus metric”) and the current conservation equation , we heuristically find that satisfies the KleinGordon equation so that (1.1) is written as
(1.2) 
where denotes the positive frequency part of the free field . However, in order to rigorously perform this procedure, one has to answer the following questions. Firstly, how to identify at a time ? Since the present state vector space is not an ordinary Hilbert space with a positive definite metric, the Hamiltonian can not be defined as a selfadjoint operator in the ordinary sense. Thus, it is far from trivial if there is a solution of quantum Heisenberg equations of motion. Secondly, is it possible to identify the positive frequency part of the operator satisfying KleinGordon equation even in an indefinite metric space? The first problem is solved by the general construction method of time evolution operator generated by a nonselfadjoint operator given by the authors [17], and as to the second one, the general definition of “positive frequency part” of a quantum field satisfying KleinGordon equation is given in Ref. [18].
Mathematically rigorous study of concrete models of QED in the Lorentz covariant gauge (see, for instance, Refs [19, 20, 18, 21]) was only given for a solvable models as far as we know. However, the model treated here, the DiracMaxwell model, is not solvable in the sense that an explicit expression of the timedependent gauge field is not easily found. Thus, the problem has to be abstractly considered, not relying on the explicit expression of the time dependent gauge field but only on the abstract existence theorem. In this paper, we establish the existence of a time evolution operator and give a definition of the “positive frequency part” of a free field safisfying KleinGordon equation in an abstract setup. Our definition of “positive frequency part” given here is different from that given in Ref [18], but results in the same consequence when applied to the concrete models. We then apply to the abstract theory to the concrete DiracMaxwell model and identify the physical Hilbert space. We also prove that the original DiracMaxwell Hamiltonian naturally defines a selfadjoint “physical Hamiltoninan” on the Hilbert space, which is essentially equivalent to the DiracMaxwell Hamiltonian in the Coulomb gauge discussed in Ref. [2].
The mathematical tools developed here would have some interests in its own right. The time evolution operator generated by unbounded, nonselfadjoint Hamiltonians has been constructed in Ref. [17]. In this paper, we further develop the theory in several aspects. Firstly, we define a general class of operators, which we will call class operators, and prove that a class operator has a time evolution for which solves the Heisenberg equation, and is times strongly differentiable in for belonging to a dense subset . Moreover, the th derivative of enjoys the natural expression in terms of a weak commutator defined in a suitable sense. Secondly, we define a more restricted class of operators, which will be called class operators, and prove that class operators has a time evolution which is analytic in . Furthermore, it would be interesting to see that the following Taylor expansion formula (2.40) remains valid even for an unbounded, nonunitary time evolution. Thirdly, the KleinGordon equation is analyzed in this abstract setup. We generally define a “free field” as a solution of the generalized KleinGordon equation, and then we prove that generalized KleinGordon equation is explicitly solved under some suitable conditions, even for an unbounded, nonunitary time evolution, even in a vector space with an indefinite metric. From this explicit solution, positive and negative frequency parts are defined unambiguously. Fourthly, abstract study with mathematical rigor makes it clear how and why the GuptaBleuler method works well and what is essentially responsible for the possibility of the method is the electromagnetic current conservation.
The paper is organized as follows. In Section 2, after the brief review of the results obtained in [17], we discuss the abstract theory of time evolution generated by nonselfadjoint Hamiltonians and give a definition of general free field to which the positive frequency part is defined. In Section 3, we intoduce the Difinition of the model and how to mathematically deal with an indefinite metric space. In Section 4, we show that the DiracMaxwell Hamiltonian is essentially selfadjoint with respect to the indefinite metric. We also show that the general theory developed so far is applicable to the DiracMaxwell Hamiltonian and to study the time evolution of quantum fields. We derive the current conservation equation and the equation of motion. In Section 5, we identify the positive frequency part of the operator by using the abstract theory of generalized free field developed in Section 4. Then, we define the physical subspace and the physical Hilbert space. We also show that the original Hamiltonian naturally defines the physical Hamiltonian which is essentially selfadjoint on the physical Hilbert space. The relation to the Coulomb gauge Hamiltonian is discussed there. Finally, we show that if the ultraviolet cutoff function of the gauge field is infrared singular, then the physical subspace has to become trivial.
2 Timeevolution Generated by Nonselfadjoint Hamiltoninans
One of the main obstacles to mathematical analysis in the Lorenz gauge is that the existence of the timeevolution operator is not trivial at all because we can not use the usual functional calculus to define the timeevolution operator if the generator is not selfadjoint. The existence of a timeevolution is established by the general theory which the authors developed in Ref. [17]. We summarize and further extend the results obtained in Ref. [17] in an abstract setup.
2.1 Existence of timeevolution
Let be a Complex Hilbert space and its inner product, and its norm. The inner product is linear in the second variable. For a linear operator in , we denote its domain (resp. range) by (resp. ). We also denote the adjoint of by and the closure by if these exist. For a selfadjoint operator , denotes the spectral measure of . The symbol denotes the restriction of a linear operator to the subspace .
Let be a selfadjoint operator on . Suppose that there is a nonnegative selfadjoint operator which is strongly commuting with . We use the notations
(2.1)  
(2.2)  
(2.3) 
Let us define a family of linear operators as follows:
Definition 2.1.
We say that a linear operator is in class if satisfies

is densely defined and closed.

and are  bounded.

There is a constant such that implies and belong to .
The set of all class operators is also denoted by the same symbol . We remark that if is in , then so is. We consider an operator
(2.4) 
with . The following Propositions summarize the results obtained in Ref. [17].
Proposition 2.1.
For each , the series:
(2.5) 
converges absolutely, where each of integrals are strong integrals. Furthermore, .
We discuss the existence of the dynamics generated by . Let
(2.6) 
Then, we have
Proposition 2.2.
For each , the vector valued functions and are strongly differentiable in . Moreover, the followings hold:
(2.7)  
(2.8) 
Further, and belong to for all .
The existence of a solution of the Heisenberg equation is ensured if in a weak sense.
Proposition 2.3.
Let . Then,

.

The operatorvalued function defined as
(2.9) is a solution of the weak Heisenberg equation:
(2.10)
In Ref. [22], a new criterion to prove the essential selfadjointness is proposed as an application of these results. We refer two results obtained there for later use.
Proposition 2.4.
Let be a symmetric operator in . If there exists a dense subspace such that for any the initial value problem
(2.11) 
has a strong solution , then, exactly one of the following (a) or (b) holds.

has no selfadjoint extension.

is essentially selfadjoint.
By using this Proposition, one readily finds
Proposition 2.5.
Let be in and symmetric. Then, is also symmetric and for the symmetric operator , exactly one of the following (a) and (b) holds:

has no selfadjoint extension.

is essentially selfadjoint.
2.2 th derivatives and Taylor expansion
In this section, we develop a general theory concerning times differentiability of the operator on a suitable subspace. Here, we take a slightly different formulation from that of Ref. [17] so that the generalization to th differentiability is easier.
To begin with, we prove a simple property of , which is not explicitly stated in Ref. [17].
Lemma 2.1.
Let . Then the mapping
(2.12) 
is strongly continuous in for all .
Proof.
Define an operator on , following Ref. [17], as
(2.13) 
Then the estimate
(2.14) 
holds for some and ([17] Lemma 3.4). From this estimate and the assumption that is in , it is straightforward to check the mapping
(2.15) 
is strongly continuous. Since can be expanded in a series converging absolutely and locally uniformly in :
(2.16) 
the limit function
(2.17) 
is also strongly continuous. ∎
Definition 2.2.
We say an operator is in class if it satisfies

is in class.

There is an operator such that
(2.18) for all .
We remark that the above operator is not unique in general. But one finds
Lemma 2.2.
Let and be an operator mentioned in Definition 2.2. Then the operator
(2.19) 
does not depend on the choice of and determined only by .
Proof.
Suppose that two operators and fulfills the condition. Then for all , we have
(2.20) 
Take arbitrary and put . Then clearly and
(2.21) 
as tends to infinity. Therefore, we have
(2.22) 
since both and are bounded. Thus one obtains
(2.23) 
Taking the closure of both sides proves the assertion. ∎
Lemma 2.3.
Let . Then the relation (2.18) remains valid for all .
Proof.
Let . Put
(2.24) 
Then, clearly and
(2.25) 
as tends to infinity for all , and the same is true for . The relation (2.18) holds for and . Thus, by the limiting argument, the assertion follows. ∎
The strong Heisenberg equation is satisfied if is in class.
Theorem 2.1.
For and the mapping
(2.26) 
is strongly continuously differentiable in and satisfies the Heisenberg equation of motion
(2.27) 
Proof.
One of the merits of the present formulation of the strong Heisenberg equation is that it is easy to extend for th differentiability.
Definition 2.3.
We define class and for inductively. That is, we say that an operator is in class if is in class and is in class. For , we write
(2.33) 
It is clear that for and that if , then , , …. We define . An operator is said to be in class if is in for all . Namely,
(2.34) 
The following Theorem is important and useful but the proof is almost trivial by induction.
Theorem 2.2.
Let is in class. Then, for all , is times strongly continuously differentiable in and
(2.35) 
In particular, if , then
(2.36) 
From Theorem 2.2, we immediately have
Theorem 2.3.
Let and . Then, there is a such that
(2.37) 
To obtain a Taylor series expansion for for , we need one more concept.
Definition 2.4.
We say that an operator is in class if

,

The operator norm
(2.38) satisfies
(2.39) 
There exists some constant such that for all , implies that belongs to .
We then arrive at the following simple result.
Theorem 2.4.
Suppose that . Then, for each , has the normconverging power series expansion formula
(2.40) 
Proof.
By Theorem 2.2, all we have to show is that the norm of the reminder term in (2.37) vanishes:
(2.41) 
Since is in class, there is a constant , which is independent of , such that implies . Choose such that implies and . Put
(2.42) 
which is finite since , and let for some . Note that is expanded in the normconverging series and is estimated as
(2.43) 
as tends to infinity. This completes the proof. ∎
2.3 Generalized KleinGordon equation and free fields
In this section, we investigate the solution of KleinGordon equation in mathematically general formulation which is suitable to the present context. Let be a complex Hilbert space and be a nonnegative selfadjoint operator on .
Definition 2.5.
A mapping is said to be free field if and only if for , belongs to class, and satisfies the differential equation:
(2.44) 
where the differentiation is the strong one.
We call this equation generalized KleinGordon equation, since in the case where and , the equation (2.44) gives the ordinary KleinGordon equation for a quantum field . As in the ordinary case, generalized KleinGordon equation can be explicitly solved, in spite of the fact that, in the present case, the time evolution is not generated by a selfadjoint Hamiltonian.
We denote
(2.45) 
Lemma 2.4.
Let be a free field. Then, for all , and
(2.46)  
(2.47) 
for all and .
Proof.
By Theorem 2.2, the generalized KleinGordon equation (2.44) is equivalent to
(2.48) 
on . At we have in particular
(2.49) 
on . Since , the right hand side belongs to , which implies that and
(2.50)  
(2.51) 
on . But again the right hand side of (2.51) belongs to , we obtain and
(2.52)  
(2.53) 
By repeating this argument, one finds that and (2.46), (2.47) hold. ∎
To solve the generalized KleinGordon equation, we introduce a wellbehaved free field:
Definition 2.6.
A free field is said to be analytic if

For all which belongs to the subspace
(2.54) is in class.

For , .

implies
(2.55) and implies
(2.56)
Theorem 2.5.
Let be an analytic free field. Then, for all , we find
(2.57) 
on .
Proof.
Take for some . Then, . Thus we have
(2.58) 
on by Theorem 2.4. By Lemma 2.4, one has for all ,
(2.59)  
(2.60) 
on . Then the Taylor expansion (2.58) becomes
(2.61) 
on . Note that as tends to infinity
(2.62) 
Thus, by the assumption (iii) in Definition 2.6, we conclude that
(2.63) 
on for all . Here, the operators and are defined through functional calculus.
Take arbitrary and put
(2.64) 
One sees that the operators and are both bounded and that converges to as tends to infinity. Hence, by the limiting argument, we find that remains valid for all . ∎
This Theorem 2.5 enables us to define positive and negative frequency parts of :
Definition 2.7.
Let be an analytic free field. We define for , on
(2.65)  
(2.66) 
and call (resp. ) positive (resp. negative) frequency part of .
The operators in the parentheses are defined though the functional calculus of selfadjoint operator . It is clear by definition that an analytic free field can be written as a sum of its positive and negative frequency parts,
(2.67) 
on the subspace .
3 Definition of the DiracMaxwell Hamiltonian in the Lorenz Gauge
In this section, we introduce the DiracMaxwell Hamiltonian quantized in the Lorenz gauge. This Hamiltonian describes a quantum system consisting of a Dirac particle under a potential and a gauge field minimally interacting with each other. We use the unit system in which the speed of light and , the Planck constant devided by , are set to be unity.
3.1 Dirac particle sector
Let us denote the mass and the charge of the Dirac particle by and , respectively. The Hilbert space of state vectors for the Dirac particle is taken to be
(3.1) 
the square integrable functions on into . The vector space here represents the position space of the Dirac particle. We sometimes omit the subscript and just write instead of when no confusion may occur. The target space realizes a representation of the four dimensional Clifford algebra accompanied by the four dimensional Minkowski vector space. The generators satisfy the anticommutation relations
(3.2) 
where the Minkowski metric tensor is given by
(3.3) 
We set , the inverse matrix of , Then we have . We assume to be Hermitian and ’s () be antiHermitian. We use the notations following Dirac:
(3.4) 
Then, s and satisfy the anticommutation relations
(3.5)  
(3.6) 
where is the Kronecker delta. The momentum operator of the Dirac particle is given by
(3.7) 
with being the generalized partial differential operator on with respect to the variable , the th component of . We write in short
The potential is represented by a Hermitian matrixvalued function on with each matrix components being Borel measurable. Note that the function naturally defines a linear operator acting in and we denote it by the same symbol . The Hamiltonian of the Dirac particle under the influence of this external potential is then given by the Dirac operator
(3.8) 
acting in , with the domain , where denotes the valued Sobolev space of order one. Let be the conjugation operator in defined by
where means the usual complex conjugation. By Pauli’s lemma [23], there is a unitary matrix satisfying
(3.9)  
(3.10) 
where for a matrix , denotes its complexconjugated matrix and the identity matrix. We assume that the potential satisfies the following conditions :
Assumption 3.1.

Each matrix component of belongs to

is ChargeParity (CP) invariant in the following sense:
(3.11) 
is essentially selfadjoint.
Hereafter, we denote the closure of , which is selfadjoint by Assumption 3.1, by the same symbol. The important remark is that the Coulomb type potential
(3.12) 
satisfies Assumption 3.1 provided that , or more concretely, if we put , the elementary charge [23].
Suppose that there are Dirac particles in the external potential . In this case, the Hilbert space should be
(3.13) 
where denotes the fold antisymmetric tensor product. The th component of
represents the position and the spinor of the th Dirac particle. For notational simplicity, we denote the positionspinor space of one electron by in what follows. We regard as a topological space with the product topology of the ordinary one on and the discrete one on . The particle Hamiltonian is then given by
(3.14) 
which is written as
(3.15) 
with denoting the generalized differential operator with respect to the th coordinate, and denoting the operators and acting in , respectively, being the mass of the th Dirac particle, and the matrixvalued multiplication operator in by the matrixvalued function , acting as