aa r X i v : . [ qu a n t - ph ] M a y Quantum Mechanics of Proca Fields
Farhad Zamani ∗ and Ali Mostafazadeh †∗ Department of Physics, Institute for Advanced Studies in BasicSciences, Zanjan 45195-1159, Iran † Department of Mathematics, Ko¸c University, Rumelifeneri Yolu,34450 Sariyer, Istanbul, Turkey
Abstract
We construct the most general physically admissible positive-definite inner product on thespace of Proca fields. Up to a trivial scaling this defines a five-parameter family of Lorentz invari-ant inner products that we use to construct a genuine Hilbert space for the quantum mechanicsof Proca fields. If we identify the generator of time-translations with the Hamiltonian, we ob-tain a unitary quantum system that describes first-quantized Proca fields and does not involvethe conventional restriction to the positive-frequency fields. We provide a rather comprehensiveanalysis of this system. In particular, we examine the conserved current density responsible forthe conservation of the probabilities, explore the global gauge symmetry underlying the conser-vation of the probabilities, obtain a probability current density, construct position, momentum,helicity, spin, and angular momentum operators, and determine the localized Proca fields. Wealso compute the generalized parity ( P ), generalized time-reversal ( T ), and generalized chargeor chirality ( C ) operators for this system and offer a physical interpretation for its PT -, C -, and CPT -symmetries.
Quantum Mechanics (QM) and Special Relativity (SR) constitute the basis for the main body ofmodern physics developed during the first half of the twentieth century. The conceptual marriageof QM and SR is Relativistic Quantum Mechanics (RQM) in which the one-particle quantum wavefunctions are identified with the solutions of an appropriate field equation. The solution space of theseequations provide the representation (Hilbert) spaces for the irreducible (projective) representationsof the Poincar´e group that in turn define the elementary particles [1, 2, 3]. One can construct theFock space associated with such a Hilbert space and define the quantum field operators as operators ∗ E-mail address: [email protected] † E-mail address: [email protected] A µ reads [7] ∂ µ F µν − M A ν = 0 , (1)where F µν := ∂ µ A ν − ∂ ν A µ , (2) M := mc ~ is the inverse of the Compton’s wave length, m ∈ R + is the mass, ∂ µ ∂ µ := η µν ∂ µ ∂ ν , and( η µν ) = ( η µν ) − := diag( − , , , ∂ ν to (1) and use F µν = − F νµ , we find that A µ satisfies the Lorentz condition: ∂ µ A µ = 0 . (3)Making use of this relation in (1) we find( ∂ ν ∂ ν − M ) A µ = 0 . (4)This means that the four components of the vector field A µ satisfy the KG equation. Note that theProca equation does not have a gauge symmetry. Therefore (3) is not a gauge choice but rather aconstraint that is to be imposed on A µ . In fact, it is not difficult to show that (4) together with(3) are equivalent to the Proca equation (1). Similarly to the case of KG fields, the presence ofthe second time-derivative in (4) is responsible for the difficulties associated with devising a soundprobabilistic interpretation for the Proca fields. 2n important motivation for the study of Proca fields is their close relationship with electro-magnetic fields. For example, the study of Proca fields may shed light on the interesting problemof the consequences of a nonzero photon rest mass which has been the subject of intensive researchover the past several decades (See the review article [9].) It might also be possible to construct afirst-quantized quantum theory of a photon by taking the zero-mass limit of that of a Proca field.This can in particular led to a resolution of the important issue of the construction of an appropriateposition operator and localized states for a photon [10].The literature on the Proca fields is quite extensive. There are a number of publications that dealwith the issue of the consistency of Proca’s theory and the difficulties associated with interactingmassive vector bosons (See [11, 12] and references therein.) It turns out that there are various waysof formulating a relativistic wave equation describing the dynamical states of a massive vector boson.The famous ones employ the equations of Proca, Duffin-Kemmer-Petiau (DKP) [13], and Weinberg-Shay-Good [14]. These are equivalent in the absence of interactions but lead to different predictionsupon the inclusion of interactions.Among the works that are directly relevant to the subject of the present article are those ofTaketani and Sakata [15], Case [16], and Foldy [17]. Using the analogy with the Maxwell equations,Taketani and Sakata [15] reduced the ten-component DKP wave function to a six-component wavefunction satisfying a Schr¨odinger equation. They made use of this representation to study theinteraction of the field with an electromagnetic field. Employing the six-component Taketani-Sakata(TS) representation, Case [16] cast the Proca equation into a form in which the positive- and negative-energy states were separately described by three-component wave functions. The latter is the so-called Foldy’s canonical representation [17] which Case used to study the non-relativistic limit ofthe Proca equation and obtained the position and spin operators acting on the TS wave functions.The approaches of Case [16] and Foldy [17] involve the use of an indefinite inner product [18] on thespace of six-component TS wave functions. This seems to be the main reason why these authorsdid not suggest any reasonable solution for the problem of the probabilistic interpretation of theirtheories. The problem of the construction of an appropriate position operator and the correspondinglocalized states for Proca fields has also been considered by various authors, e.g., [19, 20, 21, 22].But a universally accepted solution has not been given. Some other more recent articles on Procafields are [23, 24].In [25, 26, 27, 6, 28], we have employed the results obtained within the framework of Pseudo-Hermitian Quantum Mechanics (PHQM) to formulate a consistent quantum mechanics of KG fields.Here we pursue a similar approach to treat the Proca fields. The first step in this direction has beentaken by Jakubsk´y and Smejkal [29] who constructed a one-parameter family of admissible innerproducts on the space of Proca fields. In what follows we will offer a more systematic and generaltreatment of this problem. In particular we give a complete characterization of Lorentz-invariantpositive-definite inner products that render the generator of the time-translations and the helicityoperator self-adjoint. We further construct a position operator and the corresponding localized statesfor the Proca fields and examine a variety of related problems.In the remainder of this section we briefly review PHQM and give the notations and conventions3hat we use throughout the paper.PHQM [5, 30] has developed in an attempt to give a mathematically consistent formulationfor the P T -symmetric Quantum Mechanics [31]. In PHQM, a quantum system is described bya diagonalizable Hamiltonian operator H that acts in an auxiliary Hilbert space H ′ and has areal spectrum. One can show (under some general conditions) that these conditions imply that H is Hermitian with respect to a positive-definite inner product that is generally different from thedefining inner product h· , ·i of H ′ , [32]. This motivates the following definition of a pseudo-Hermitianoperator [33]: H is a pseudo-Hermitian operator if there is a linear, invertible, Hermitian (metric)operator η : H ′ → H ′ satisfying H † = ηHη − . This condition is equivalent to the requirement that H be self-adjoint with respect to the (possibly indefinite) inner product hh· , ·ii η := h· , η ·i . It turnsout that for a given pseudo-Hermitian operator H , the metric operator η is not unique. If we makea particular choice for η , we say that H is η -pseudo-Hermitian .A proper subset of pseudo-Hermitian operators is the set of quasi-Hermitian operators [34].A quasi-Hermitian operator H is a pseudo-Hermitian operator that is η + -pseudo-Hermitian for apositive-definite metric operator η + , i.e., H † = η + Hη − . (5)This means that H is Hermitian (self-adjoint) with respect to the positive-definite inner product hh· , ·ii η + := h· , η + ·i . (6)In particular H is a diagonalizable operator with a real spectrum and can be mapped to a Hermitianoperator via a similarity transformation [32].A quasi-Hermitian Hamiltonian operator defines a quantum system with a consistent probabilisticinterpretation provided that one constructs the physical Hilbert space H of the system using the innerproduct hh· , ·ii η + and identifies the observables of the theory with the self-adjoint operators acting in H , [34, 5]. Although PHQM employs quasi-Hermitian Hamiltonian operators, the study of the pseudo-Hermitian Hamiltonians has proved to be essential in understanding the role of anti-linear symmetriessuch as
P T . As shown in Ref. [35], given a diagonalizable pseudo-Hermitian Hamiltonian H , onemay introduce generalized parity ( P ), time-reversal ( T ) and charge or chirality ( C ) operators and es-tablish the C -, PT -, and CPT -symmetries of H that would respectively generalize C , P T , and
CP T symmetries [36] of
P T -symmetric quantum mechanics. We recall that H is P -pseudo-Hermitianand T -anti-pseudo-Hermitian, P and T are respectively linear and anti-linear invertible operators,and C is a linear involution ( C = 1) satisfying C = η − P , [35]. Next, we give our notations and conventions. H is the Cauchy completion of the inner product space obtained by endowing the span of the eigenvector of H (in H ′ ) with the inner product hh· , ·ii η + . See also [37]. A simple consequence of the latter relation is that the
CP T -inner product of [36] is a particular example of theinner products (6) of [32, 38, 39]. a = ( a , a ) stands for a four-vector a µ , Greek indices take on the values0 , , ,
3, Latin indices take on 1 , ,
3, and ε ijk denotes the totally antisymmetric Levi-Civita symbolwith ε = 1. We employ Einstein’s summation convention over repeated indices and use σ and λ to denote the 2 × × σ together with the Paulimatrices σ = ! , σ = − ii ! , σ = − ! , (7)form a basis of the vector space of 2 × { σ , σ } is a maximal set ofcommuting matrices. We denote their common eigenvectors by e + := ! and e − := ! , wherethe labels correspond to the eigenvalues of σ .Similarly, λ together with λ = , λ = − i i , λ = − , λ = ,λ = − i i , λ = , λ = − i i , λ = 1 √ − , (8)form a basis for the vector space of 3 × { λ , λ , λ } is a set of maximallycommuting matrices whose common eigenvectors we denote by e +1 := , e − := , and e := . Again the labels 0 and ± λ .Using the bases { σ i } and { λ j } we can construct the basis { Σ m } for the vector space of 6 × m ∈ { , , , · · · , } Σ m := σ i ⊗ λ j , if m = i + 4 j. (9)The operators Σ , Σ , Σ , Σ , Σ , and Σ form a maximal set of commuting operators withcommon eigenvectors e ǫ,s := e ǫ ⊗ e s , (10)where ǫ ∈ { + , −} and s ∈ {− , , +1 } .The organization of the paper is as follows. In Section 2, we present a six-component formulationof the Proca equation, establish its relation with pseudo-Hermiticity, and construct a positive-definitemetric operator and the corresponding inner product. In Section 3, we compute the generalizedparity P , time-reversal T , and charge or chirality C operators and elaborate on the C -, PT -, and CPT -symmetries of the Hamiltonian. In Section 4, we derive the expression for the most generalphysically admissible positive-definite inner product on the solution space of the Proca equation anddemonstrate the unitary-equivalence of the representation obtained by the corresponding Hilbertspace and the generator of time-translations as the Hamiltonian with the Foldy representation. InSection 5, we obtain and explore the properties of a conserved current density that supports the5onservation of the probabilities. In Section 6, we introduce a position basis and the associatedposition wave functions for the Proca fields. In Section 7, we discuss the position, spin, helicity,momentum and angular momentum operators, and construct the relativistic localized states for theProca fields. In Section 8, we compute the probability current density for the spatial localizationof a Proca field. In Section 9, we study the gauge symmetry associated with the conservation ofprobabilities. Finally in Section 10, we present a summary of our main results and discuss thedifferences between our approach of finding the most general admissible inner product and that ofRef. [29].
To begin our investigation, we briefly formulate the covariant dynamical theory of Proca fields andreview the physical polarization and helicity states.Consider a Proca field A µ such that for all x ∈ R and µ ∈ { , , , } , R R d x | A µ ( x , x ) | < ∞ .Then we can express (4) as a dynamical equation in the Hilbert space L ( R ) ⊕ L ( R ) ⊕ L ( R ) ⊕ L ( R ), namely ¨ A ( x ) + DA ( x ) = 0 , (11)where a dot denotes a x -derivative, for all µ ∈ { , , , } and x ∈ R the functions A µ ( x ) : R → C are defined by A µ ( x )( x ) := A µ ( x , x ) , ∀ x ∈ R , and D : L ( R ) → L ( R ) is the Hermitian operator:[ Df ]( x ) := [ M − ∇ ] f ( x ) , ∀ f ∈ L ( R ) . (12)Clearly, for all µ ∈ { , , , } and x ∈ R , A µ ( x ) belongs to L ( R ), and D is a positive-definiteoperator with eigenvalues ω k := k + M and the corresponding eigenvectors φ k ( x ) := (2 π ) − / e i k · x = h x | k i , where k ∈ R , k := | k | , and h·|·i denotes the inner product of L ( R ).It is important to observe that a solution of (11) is a Proca field provided that for all x ∈ R thefunctions A ( x ) fulfill the Lorentz condition that we can express as the constraint: L [ A ( x )] := ˙ A ( x ) + i K · A ( x ) = 0 , (13)where for all f ∈ L ( R ), ( K f )( x ) := − i ∇ f ( x ), and “ · ” is the usual dot product.The Lorentz condition (13) implies that there are only three independent components of A ; usingthis condition we can express A ( x ) and ˙ A ( x ) in terms of A ( x ) and ˙ A ( x ). This means that theinitial data for the Proca equation (11) is given by (cid:16) A ( x ) , ˙ A ( x ) (cid:17) for some initial value x ∈ R of x . We shall however employ a manifestly covariant approach and treat all components of the field A ( x ) on an equal footing. As a result we define the complex vector space ( V ) of solutions of theProca equation (11) as V := (cid:8) A := ( A , A ) | ∀ x ∈ R , [ ∂ + D ] A µ ( x ) = 0 , L [ A ( x )] = 0 (cid:9) . (14)6t is not difficult to show that the Lorentz condition (13) may be imposed in the form of a constrainton the initial data for the dynamical equation (11), namely L [ A ( x )] = L [ ˙ A ( x )] = 0 . (15)This in turn implies V = n A := ( A , A ) | ∀ x ∈ R , [ ∂ + D ] A µ ( x ) = 0 , and ∃ x ∈ R , L [ A ( x )] = L [ ˙ A ( x )] = 0 o . (16)We can express the solution of (11) in terms of ( A µ ( x ) , ˙ A µ ( x )) according to [26, 40] A µ ( x ) = cos[( x − x ) D / ] A µ ( x ) + sin[( x − x ) D / ] D − / ˙ A µ ( x ) . (17)Note that here and throughout this paper we use the spectral resolution of D to define its powers, D α := R R d k ( k + M ) α | k ih k | for all α ∈ R .Next, consider the plane wave solutions of (4): A µǫ ( k , σ ; x ) = N ǫ, k a µǫ ( k , σ ) e − iǫω k x φ k ( x ) , (18)where N ǫ, k are normalization constants (Lorentz scalars), and a µǫ ( k , σ ) denotes a set of normalizedfour-dimensional polarization vectors fulfilling the Lorentz condition, i.e., a ǫ µ ( k , σ ) a µǫ ( k , σ ′ ) = δ σσ ′ , k µ a µǫ ( k , σ ) = 0 . (19)These relations show that the Proca field (18) has only three physical polarization states ( σ ∈{ , , } ). In a fixed reference frame in which the plane wave has momentum k , we choose to workwith a pair of transverse polarization vectors : [3, 41] a ǫ ( k ,
1) = (0 , a ǫ ( k , , a ǫ ( k ,
2) = (0 , a ǫ ( k , , (20)whose space-like components, a ǫ ( k , σ ) with σ = 1 ,
2, are normalized vectors perpendicular to k . Asa longitudinal polarization vector we choose [3, 41] a ǫ ( k ,
3) = (cid:18) k M , k M k k (cid:19) , k = ǫ ω k , (21)which together with (20) form an orthonormal set. These polarization vectors also fulfil the com-pleteness relation [3, 41]: X σ =1 a µǫ ( k , σ ) a νǫ ( k , σ ) = η µν + 1 M k µ k ν . (22) The ¨ A ( x ) in L [ ˙ A ( x )] stands for − DA ( x ). The longitudinal polarization vector used in Ref. [29] is not normalized. Note also that, unlike the authors ofRef. [29], we do not fix the transverse polarization vectors a ǫ ( k , σ ) , σ = 1 ,
2. Although we fix a reference frame in whichtheir time-like components are zero, their space-like components are still arbitrary. Instead of fixing these vectors,we only fix the longitudinal polarization vector (21) and make use of equations (19) and (22) to do the necessarycalculations throughout this paper.
7t turns out that the use of the circular polarization vectors , u ǫ, ( k ) := a ǫ ( k , , u ǫ, ± ( k ) := 1 √ a ǫ ( k , ± i a ǫ ( k , , (23)simplifies many of the calculations. This is because their spatial components are eigenvectors of thehelicity operator: h = K · S | K | = ˆ K · S = i | K | − K K K − K − K K , (24)where K =: ( K , K , K ), ˆ K := K / | K | , and S := λ , S := − λ , S := λ , (25)are the angular momentum operators in the spin-one representation of the rotation group [42]. Theysatisfy S i S j S k + S k S j S i = δ ij S k + δ kj S i , [ S i , S j ] = iǫ ijk S k . (26)A simple consequence of these relations is the identity h = h which we shall make use of in thesequel.We can also construct the basic (plane wave) solutions of (4) that have a definite helicity. Theseare given by A µǫ,h ( k ; x , x ) = N ǫ,h ( k ) u µǫ,h ( k ) e − iǫω k x φ k ( x ) , ǫ = ± , h = − , , , (27)where N ǫ,h ( k ) are normalization constants. In analogy with electromagnetism, we express the antisymmetric tensor F µν in terms of the E and B fields [43]: F ij =: ε ijm B m , F i =: E i . (28)Then the Proca equation (1) takes the form B = ∇ × A , ˙ A = − E − ∇ A ,A = − M− ∇ · E , ˙ E = M A + ∇ × B . (29)Next, we let ˜ H := { Y : R → C | ≺ Y , Y ≻ < ∞} denote the Hilbert space of vector fields, where forall Y , Z : R → C , ≺ Y , Z ≻ := R R d x Y ( x ) ∗ · Z ( x ). If we eliminate B and A , we can express (29)as a set of dynamical equations in the Hilbert space ˜ H , namely˙ A ( x ) = − E ( x ) + M− D E ( x ) , ˙ E ( x ) = D A ( x ) + D A ( x ) , (30)where D : ˜ H → ˜ H is the Hermitian operator:[ D Y ]( x ) := (cid:2) [( K · S ) − K ] Y (cid:3) ( x ) = ∇ ( ∇ · Y ( x )) , (31)8 ∈ ˜ H is arbitrary, and we have used the identities[ K Y ]( x ) = −∇ Y ( x ) , [( K · S ) Y ] ( x ) = ∇ × Y ( x ) , (32)which also imply K · [( K · S ) Y ] = 0 , K · [( K · S ) Y ] = 0 . (33)Next we express (30) as the Schr¨odinger equation (also known as the Taketani-Sakata equation[15, 16]) i ~ ˙Ψ( x ) = H Ψ( x ) , (34)where for all x ∈ R the state vector Ψ and the Hamiltonian H are defined byΨ( x ) := A ( x ) − iγ E ( x ) A ( x ) + iγ E ( x ) ! , (35) H := ~ H H − H − H ! , (36) γ ∈ R − { } is an arbitrary constant having the dimension of length, and H and H are the followingHermitian operators that act in ˜ H . H := γ ( D + D ) + γ − M− ( M − D ) = γ ( M + ( K · S ) ) + γ − M− ( D − ( K · S ) ) , (37) H := γ ( D + D ) − γ − M− ( M − D ) = γ ( M + ( K · S ) ) − γ − M− ( D − ( K · S ) ) . (38)Note that one can invert the first equation in (30) (or take a x -derivative of the second equation in(30) and use ¨ E = − D E ) to obtain E ( x ) = − D − (cid:2) M + ( K · S ) (cid:3) ˙ A ( x ) . (39)In view of (35) and (39), the six-component state vector Ψ is completely determined in terms of A and ˙ A . This means that the space of the state vectors Ψ is isomorphic (as a vector space) to thespace of the initial conditions ( A ( x ) , ˙ A ( x )). Because of the linearity of the Proca equation, thelatter is also isomorphic to the vector space of Proca fields. The six-component vectors Ψ( x ) belong to C ⊗ ˜ H . If we endow the latter with the inner product h· , ·i defined by h ξ, ζ i := X i =1 ≺ ξ i , ζ i ≻ , (40) Similarly to the two-component representation of the KG fields [25], the choice of the six-component state vector(35) is not unique. Its general form is Ψ g ( x ) = g ( x )Ψ where g ( x ) ∈ GL (6 , C ). (30) is equivalent to the Schr¨odingerequation i ~ ˙Ψ g ( x ) ( x ) = H g ( x ) Ψ g ( x ) ( x ) where H g ( x ) := g ( x ) Hg ( x ) − + i ~ ˙ g ( x ) g ( x ) − . The arbitrariness in thechoice of g ( x ) is related to a GL (6 , C ) gauge symmetry of the six-component formulation of the Proca fields. We willtake g ( x ) to be the identity matrix. This is a partial gauge-fixing, because we do not fix γ . Changing γ correspondsto a gauge transformation associated with a GL (1 , R ) subgroup of GL (6 , C ), namely Ψ γ → Ψ γ ′ = g ( γ, γ ′ )Ψ γ where g ( γ, γ ′ ) = γ γ + γ ′ γ − γ ′ γ − γ ′ γ + γ ′ ! ⊗ λ . This gauge symmetry has no physical significance, and as we shall see our finalresults will not depend on γ . ξ = ξ ξ ! , ζ = ζ ζ ! ∈ C ⊗ ˜ H , and denote the corresponding Hilbert space, namely ˜ H⊕ ˜ H ,by H ′ , we can view Ψ( x ) as elements of H ′ and identify H with a linear operator acting in H ′ . Onecan easily check that H : H ′ → H ′ is not a Hermitian operator, but it satisfies H † = Σ H Σ − ,i.e., it is Σ -pseudo-Hermitian. This implies that H is self-adjoint with respect to the inner product hh· , ·ii Σ := h· , Σ ·i on C ⊗ ˜ H . This in turn induces the following inner product on the space (14) ofthe Proca fields. (( A, A ′ )) Σ3 := g γ hh Ψ( x ) , Ψ ′ ( x ) ii Σ = g γ h Ψ( x ) , Σ Ψ ′ ( x ) i = ig h ≺ A , ( ˙ A ′ + ∇ A ′ ) ≻ − ≺ ( ˙ A + ∇ A ) , A ′ ≻ i , (41)where g ∈ R + is a constant. Because of Σ -pseudo-Hermiticity of H , this inner product, which issometimes called the Proca inner product , is invariant under the time-evolution generated by H , butit is obviously indefinite.The dynamical invariance of (41) is associated with a conserved current density, namely [43, 24] J µ Σ3 ( x ) := ig [ A ν ( x ) ∗ F νµ ( x ) − F νµ ( x ) ∗ A ν ( x )] . (42)This is the spin-one analog of the KG current density. We can express the Proca inner product ina manifestly covariant form in terms of the current density (42). But because this inner productis indefinite, it cannot be used to make the solution space of the Proca equation into a genuineinner product space. Again in analogy to the case of KG fields, one may pursue the approach ofindefinite-metric quantum theories [18] and restrict to the subspace of positive-energy solutions (e.g.see [44]). But this scheme has the same difficulties as the corresponding treatment of the KG fields. Next, we will use the positive-definiteness of D to show that H is a diagonalizable operatorwith a real spectrum. This suggests that it is η + -pseudo-Hermitian for a positive-definite metricoperator η + [32]. Therefore, H is Hermitian with respect to the positive-definite inner product: hh· , ·ii η + := h· , η + ·i . The construction of η + requires the solution of the eigenvalue problem for H and H † .The eigenvalue problem for the Hamiltonian (36) may be easily solved. H has a symmetrygenerated by Λ := σ ⊗ h , (43)which is the helicity operator in the six-component representation of the Proca fields. The simulta-neous eigenvectors of H and Λ are given byΨ ǫ,h ( k ) := 12 ( r − k + ǫr k ) u ǫ,h ( k ) − γr − k k u ǫ,h ( k )( r − k − ǫr k ) u ǫ,h ( k ) + γr − k k u ǫ,h ( k ) ! φ k , (44) H Ψ ǫ,h ( k ) = E ǫ ( k )Ψ ǫ,h ( k ) , ΛΨ ǫ,h ( k ) = h Ψ ǫ,h ( k ) , (45) For example, although the inner product (41) restricted to the positive-energy solutions is positive definite, J Σ3 ( x )that should correspond to the probability density is not generally positive-definite even for positive-energy fields. Hereafter we will omit ⊗ wherever there is no risk of confusion. ǫ ∈ {− , } , h ∈ {− , , +1 } , k ∈ R , k := | k | , ω k := √ k + M , r k := √ γω k , E ǫ ( k ) := ǫ ~ ω k ,u ǫ,h ( k ) are circular polarization vectors (23), and φ k are eigenvectors of D corresponding to theeigenvalues ω k . The eigenvectors Ψ ǫ,h ( k ) together withΦ ǫ,h ( k ) := 12 ( r k + ǫr − k ) u ǫ,h ( k ) − ǫγr − k k u ǫ,h ( k )( r k − ǫr − k ) u ǫ,h ( k ) − ǫγr − k k u ǫ,h ( k ) ! φ k , (46)form a complete biorthonormal system for the Hilbert space. This means that h Ψ ǫ,h ( k ) , Φ ǫ ′ ,h ′ ( k ′ ) i = δ ǫ,ǫ ′ δ h,h ′ δ ( k − k ′ ) , X ǫ = ± X h =0 , ± Z R d k | Ψ ǫ,h ( k ) ih Φ ǫ,h ( k ) | = Σ , (47)where h· , ·i stands for the inner product of H ′ , and for ξ, ζ ∈ H ′ , | ξ ih ζ | is the operator defined by | ξ ih ζ | χ := h ζ , χ i ξ , for all χ ∈ H ′ . Similarly, one can check that indeed Φ ǫ,h ( k ) are simultaneouseigenvectors of H † and Λ † = Λ with the same eigenvalues, H † Φ ǫ,h ( k ) = E ǫ ( k )Φ ǫ,h ( k ), ΛΦ ǫ,h ( k ) = h Φ ǫ,h ( k ), and that H and Λ have the following spectral resolutions H = X ǫ = ± X h =0 , ± Z R d k E ǫ ( k ) | Ψ ǫ,h ( k ) ih Φ ǫ,h ( k ) | , Λ = X ǫ = ± X h =0 , ± Z R d k h | Ψ ǫ,h ( k ) ih Φ ǫ,h ( k ) | . Another remarkable property of the biorthonormal system { Ψ ǫ,h ( k ) , Φ ǫ,h ( k ) } is that h Ψ ǫ,h ( k ) , Ψ ǫ ′ ,h ′ ( k ′ ) i = h Ψ ǫ ′ ,h ′ ( k ′ ) , Ψ ǫ,h ( k ) i = ǫ ǫ ′ h Φ ǫ,h ( k ) , Φ ǫ ′ ,h ′ ( k ′ ) i . (48)Using the properties of the polarization vectors, i.e., Eqs. (19) and (22), we can compute thepositive-definite metric operator, η + : H ′ → H ′ , associated with the biorthonormal system { Ψ ǫ,h ( k ) , Φ ǫ,h ( k ) } . The result is η + = X ǫ = ± X h =0 , ± Z R d k | Φ ǫ,h ( k ) ih Φ ǫ,h ( k ) | = D − / H H H H ! . (49)The inverse of η + has the form η − = X ǫ = ± X h =0 , ± Z R d k | Ψ ǫ,h ( k ) ih Ψ ǫ,h ( k ) | = D − / H − H − H H ! . (50)Now we are in a position to compute hh· , ·ii η + . For all ξ, ζ ∈ H ′ , we let ξ i , ζ i , ξ ± , ζ ± ∈ ˜ H bedefined by ξ ξ ! := ξ, ζ ζ ! := ζ , ξ ± := ξ ± ξ , ζ ± := ζ ± ζ . (51)Then in view of (49), (37), (38), and (40), hh ξ, ζ ii η + = 12 (cid:20) γ ≺ ξ + , D − / [ M + ( K · S ) ] ζ + ≻ + 1 γ M ≺ ξ − , D − / [ D − ( K · S ) ] ζ − ≻ (cid:21) . (52)11f we view H ′ as a complex vector space and endow it with the inner product (52), we obtain a newinner product space whose Cauchy completion yields a Hilbert space which we denote by K .Next, for all x ∈ R we define U x : V → H ′ according to U x A := 12 r κγ M Ψ( x ) , ∀ A ∈ V , (53)where κ ∈ R + is a fixed but arbitrary constant. We can use U x to endow the complex vector space V of Proca fields with the positive-definite inner product(( A, A ′ )) := hh U x A, U x A ′ ii η + = κ γ M hh Ψ( x ) , Ψ ′ ( x ) ii η + . (54)Because η + does not depend on x , the inner product hh· , ·ii η + is invariant under the dynamicsgenerated by H , [33]. This in turn implies that the right-hand side of (54) should be x -independent.In order to see this, we substitute (35) and (52) in (54) and use (30) and (31) to derive(( A, A ′ )) = κ M (cid:2) ≺ A ( x ) , D − / [ M + ( K · S ) ] A ′ ( x ) ≻ + M− ≺ ( ˙ A ( x ) + i K A ( x )) , D − / [ D − ( K · S ) ]( ˙ A ′ ( x ) + i K A ′ ( x )) ≻ i = κ M h ≺ A ( x ) , D − / ˙ E ′ ( x ) ≻ − ≺ E ( x ) , D − / ˙ A ′ ( x ) ≻ i , (55)where E ( x ) = − ˙ A ( x ) − i K A ( x ). We can use (11) or equivalently (30) to check that the x -derivative of the right-hand side of (55) vanishes identically. Therefore, (55) provides a well-definedinner product on V . Endowing V with this inner product and (Cauchy) completing the resultinginner product space we obtain a separable Hilbert space which we shall denote by H . This is thephysical Hilbert space of the relativistic quantum mechanics of the Proca fields.The inner product (55) is identical with an inner product obtained in [29]. We will show inSection 4 that it is a special example of a larger class of invariant inner products and that it has thefollowing appealing properties:1. It is not only positive-definite but relativistically invariant.
2. Its restriction to the subspace of positive-frequency Proca fields coincides with the restrictionof the indefinite Proca inner product (41) to this subspace.As seen from (54), the operator U x for any value of x ∈ R is a unitary operator mapping H to K . Following [26, 27] we can use this unitary operator to define a Hamiltonian operator h acting in H that is unitary-equivalent to H . Let x ∈ R be an arbitrary initial x , and h : H → H be definedby h := U − x H U x . (56)Then, using (29), (35) – (38) and (53), we can easily show that for any A ∈ V ,h A = i ~ ˙ A, (57) We give a manifestly covariant expression for this inner product in Section 5. A is the element of V defined by: ˙ A ( x ) := ddx A ( x ), for all x ∈ R . As discussed in [26] forthe case of KG fields, one must not confuse (57) with a time-dependent Schr¨odinger equation givingthe x -dependence of A ( x ). This equation actually defines the action of the operator h on the field A . h generates a time-evolution, through the Schr¨odinger equation i ~ ddx A x = h A x , (58)that coincides with a time-translation in the space V of the Proca fields; if A = A x is the ini-tial value for the one-parameter family of elements A x of V , then for all x , x ′ ∈ R , A x ( x ′ ) =( e − i ( x − x )h / ~ A )( x ′ ) = A ( x ′ + x − x ). Furthermore, using the fact that U x is a unitary operatorand that H is Hermitian with respect to the inner product (52) of K , we can infer that h is Hermitianwith respect to the inner product (54) of H . This shows that time-translations correspond to unitarytransformations of the physical Hilbert space H .Next, we define U : H → H ′ and H ′ : H ′ → H ′ by U := ρ U x (59) H ′ := U h U − = ρHρ − , (60)where ρ is the unique positive square root of η + , i.e., ρ := √ η + . It is not difficult to see that ρ = 12 M √ γ ρ + ρ − ρ − ρ + ! , ρ − = 12 M √ γ ρ + − ρ − − ρ − ρ + ! , (61)where ρ ± : ˜ H → ˜ H are Hermitian operators given by ρ ± := ( γ M ∓ D / − M D − / ] h + γ M D − / ± D / , (62)and h is the helicity operator (24). We can check that the operator ρ viewed as mapping K to H ′ isa unitary transformation; using ρ † = ρ = √ η + , we have h ρξ, ρζ i = hh ξ, ζ ii η + for all ξ, ζ ∈ K . This inturn implies that U : H → H ′ is also a unitary transformation, hU A, U A ′ i = (( A, A ′ )) , ∀ A, A ′ ∈ H , (63)and that H ′ must be a Hermitian Hamiltonian operator acting in H ′ .We can compute H ′ by substituting (61), (62) and (36) – (38) in (60). This yields H ′ = ~ √ D −√ D ! = ~ √ D Σ , (64)which is manifestly Hermitian with respect to the inner product h· , ·i of H ′ . The Hamiltonian H ′ is precisely the Foldy Hamiltonian [17, 16]. Here we obtained it by a systematic application of themethods of PHQM [5]. 13ext, we derive the explicit form of U and its inverse. Using (59), (61), (62), (53), and (35), wehave for all A ∈ H , U A = 12 r κ M U A ( x ) − i U − E ( x ) U A ( x ) + i U − E ( x ) ! , (65)where the operator U : ˜ H → ˜ H and its inverse are given by U = [ D / − M D − / ] h + M D − / , U − = [ D − / − M− D / ] h + M− D / . (66)The inverse U − of U is also easy to calculate. Let ξ ∈ H ′ be a six-component vector (as in (51)) withcomponents ξ and ξ . Then in view of (65), U − ξ is the Proca field A ∈ H satisfying the followinginitial conditions: A ( x ) = D − / √ M κ K · ( ξ − ξ ) , ˙ A ( x ) = − i D / √ M κ K · ( ξ + ξ ) , (67) A ( x ) = r M κ U − ( ξ + ξ ) , ˙ A ( x ) = − i r M κ D / U − ( ξ − ξ ) , (68)where we have made use of equations (15), (30), (33), (35), (53), (61), and (62). By virtue of (17),for all x ∈ R , we have A ( x ) = D − / √ M κ h e − i ( x − x ) D / K · ξ − e i ( x − x ) D / K · ξ i , (69) A ( x ) = r M κ U − h e − i ( x − x ) D / ξ + e i ( x − x ) D / ξ i . (70)In complete analogy with the case of KG fields [27], we find that the pairs ( H , h), ( K , H ), and ( H ′ , H ′ )are unitarily equivalent. Therefore they represent the same quantum system. P T , C and C P T -symmetries of Proca Fields
According to [35], the generalized parity ( P ), time-reversal ( T ), and charge grading or chirality ( C )operators for the six-component Proca fields are given by P := X ǫ = ± X σ =1 Z R d k σ ǫ,σ ( k ) | Φ ǫ,σ ( k ) ih Φ ǫ,σ ( k ) | , (71) T := X ǫ = ± X σ =1 Z R d k σ ǫ,σ ( k ) | Φ ǫ,σ ( k ) i ⋆ h Φ ǫ,σ ( k ) | , (72) C := X ǫ = ± X σ =1 Z R d k σ ǫ,σ ( k ) | Ψ ǫ,σ ( k ) ih Φ ǫ,σ ( k ) | , (73) Transformation (65) is known as the Foldy’s transformation, although it differs slightly from the expression givenby Foldy [17] and Case [16], namely U Foldy := Σ U . This difference turns out not to have any effect on the definitionof the physical observables such as the position operator. It is interesting to see that the arbitrary parameter γ , introduced in the six-component formulation of the Procaequation, does not appear in (64) and (65). σ ǫ,σ ( k ) are arbitrary signs ( ± ), and ⋆ is the complex-conjugation operator defined, for allcomplex numbers z and state vectors Ψ , Φ, by ⋆z := z ∗ and ( ⋆ h Φ | ) | Ψ i := ⋆ h Φ | Ψ i = h Ψ | Φ i . Notethat we use the biorthonormal basis { Ψ ǫ,σ ( k ) , Φ ǫ,σ ( k ) } to define P , T , and C . This basis can beobtained by replacing the circular polarization vectors u ǫ,h ( k ) by the linear polarization vectors a ǫ ( k , σ ) in (44) and (46).Clearly there is an infinity of choices for the signs σ ǫ,σ ( k ) each leading to a different P , T , and C .Following [27] we will adopt the natural choice associated with the label ǫ appearing in the expressionfor the eigenvalues and eigenvectors of H . This yields P := X ǫ = ± X h =0 , ± Z R d k ǫ | Φ ǫ,h ( k ) ih Φ ǫ,h ( k ) | , (74) T := X ǫ = ± X h =0 , ± Z R d k ǫ | Φ ǫ,h ( k ) i ⋆ h Φ ǫ, − h ( k ) | , (75) C := X ǫ = ± X h =0 , ± Z R d k ǫ | Ψ ǫ,h ( k ) ih Φ ǫ,h ( k ) | . (76)The consequences of these relations are identical with those obtained in [27] for the KG fields. Becauseof the close analogy with the case of KG fields, here we omit the details and give a summary of therelevant results.Substituting (44) and (46) in (74) – (76) yields P = Σ , T = Σ ⋆, PT = ⋆. (77)Hence, the PT -symmetry of the Hamiltonian (36) means that it is a real operator. Note that P = T = Σ and PT = P · T , which is a direct consequence of (48), [35]. Similarly, we find C = D − / H H − H − H ! = ~ − D − / H = H √ H , (78)which in view of (49) is consistent with the identity C = η − P (equivalently η + = PC ) [35].According to (78), C is a Z -grading operator for the Hilbert space that splits it into the spans ofthe eigenvectors of H with positive and negative eigenvalues, respectively.We can use the unitary operator ρ : K → H ′ to express the generalized parity, time-reversal, andchirality operators in the Foldy representation, namely P ′ := ρ P ρ − , T ′ := ρ T ρ − , and C ′ := ρ C ρ − .The symmetry generators P ′ T ′ and C ′ that commute with the Foldy Hamiltonian H ′ have the form P ′ T ′ = PT = ⋆, C ′ = H ′ √ H ′ = Σ . (79)Clearly, the P ′ T ′ -symmetry of H ′ is related to the fact that H ′ is a real operator, and the C ′ -symmetryof H ′ arises because it is proportional to C ′ . Note that P , T and the metric operator η + depend on the choice of the biorthonormal system, while C isindependent. In [45] this fact has not been considered in the factorization of the metric operator. U : H → H ′ to define the generalized parity, time-reversal, and chirality operators for the ordinary Proca fields A ∈ H . This yieldsP := U − P ′ U , T := U − T ′ U , C := U − C ′ U . (80)Using Eqs. (65) – (68), and (17), we may obtain explicit expressions for P and T. Here we give thecorresponding expressions for PT and C, that actually generate symmetries of the Hamiltonian h.The result is, for all A ∈ H and x ∈ R ,[PT A ]( x ) = ( − A ( − x ) ∗ , A ( − x ) ∗ ) , (81)[C A ] ( x ) = iD − / ˙ A ( x ) =: A c ( x ) . (82)These relations imply that, similarly to the case of KG field [27], PT is just the ordinary time-reversal operator [17], and “ the PT -symmetry of h means that the order in which one performs atime-translation and a time-reversal transformation on a Proca field is not important .” Furthermore,chirality operator acting in H is simply given by C = iD − / ∂ , which is a Lorentz scalar [6]. RecallingC A = A , we observe that C : V → V is an involution, C = 1. Therefore, we can use it to split V into the subspaces V ± of ± -energy Proca fields according to V ± := { A ± ∈ V| C A ± = ± A ± } . Clearly,for any A ∈ V , we can use C to introduce the corresponding ± -energy components: A ± := 12 ( A ± C A ) ∈ V ± . (83)Clearly A = A + + A − , so V = V + ⊕ V − . Restricting the inner product (55) to V ± (and Cauchycompleting the resulting inner product spaces) we obtain Hilbert subspaces H ± of H .Next we recall that C ′ is a Hermitian operator acting in H ′ . Thus, in view of (80), and (63), wehave (( A, C A ′ )) = ((C A, A ′ )). Therefore, C : H → H is a Hermitian involution and for all A ± ∈ V ± ,(( A + , A − )) = 0. This in turns implies H = H + ⊕ H − . The generalized chirality operator C is actuallythe grading operator associated with this orthogonal direct sum decomposition of H . In other words,similarly to the case of KG fields [27], C is a Hermitian involution that decomposes the Hilbert spaceinto its ± -energy subspaces. As a result, “ the C -symmetry of h means that the energy of a free Procafield does not change sign under time-translations .” The metric operator η + and the corresponding invariant positive-definite inner product depend on thechoice of the biorthonormal system { Ψ ǫ,h ( k ) , Φ ǫ,h ( k ) } . The most general invariant positive-definiteinner product corresponds, therefore, to the most general biorthonormal system that consists ofthe eigenvectors of H and H † . In the construction of { Ψ ǫ,h ( k ) , Φ ǫ,h ( k ) } we have chosen a set ofeigenvectors of H that also diagonalize the helicity operator Λ. As a result, both H and Λ are η + -pseudo-Hermitian. This in turn is equivalent to the condition that H and Λ are among the This marks its similarity to the chirality operator γ for spin 1/2 fields. The most general positive-definite operator ˜ η + that renders both H and Λ ˜ η + -pseudo-Hermitianhas the form ˜ η + := A † η + A where A is an invertible linear operator commuting with H and Λ [39, 35].We can express A as A = X ǫ = ± X h =0 , ± Z R d k α ǫ,h ( k ) | Ψ ǫ,h ( k ) ih Φ ǫ,h ( k ) | , (84)where α ǫ,h ( k ) are arbitrary nonzero complex numbers. This implies˜ η + := A † η + A = X ǫ = ± X h =0 , ± Z R d k | α ǫ,h ( k ) | | Φ ǫ,h ( k ) ih Φ ǫ,h ( k ) | . (85)Substituting (44) and (46) in (85), and carrying out the necessary calculations, we find˜ η + = (cid:2) L ++ M + ( L + − − L − ) σ − L N (cid:3) h + (cid:2) L − + M + L −− σ (cid:3) h + L N + L − σ , (86)where L ǫ ± := 14 Z d k (cid:2) | α + , ( k ) | + ǫ | α + , − ( k ) | ± | α − , ( k ) | ± ǫ | α − , − ( k ) | (cid:3) | k ih k | , (87) L ± := 12 Z d k (cid:2) | α + , ( k ) | ± | α − , ( k ) | (cid:3) | k ih k | , (88) M := D − / γ γ D + 1 γ D − γ D − γ D + 1 ! , (89) N := D − / γ M γ M + D γ M − Dγ M − D γ M + D ! . (90)Having computed ˜ η + we can easily obtain the corresponding inner product according to hh· , ·ii ˜ η + := h· , ˜ η + ·i . This yields, for all ξ, ζ ∈ H ′ , hh ξ, ζ ii ˜ η + = 12 (cid:2) γ ≺ ξ + , Θ + , +1 ζ + ≻ + γ − ≺ ξ − , Θ + , − ζ − ≻ (cid:3) + ≺ ξ , Θ − , ζ ≻ − ≺ ξ , Θ − , ζ ≻ , (91) where the operators Θ ǫ,h : ˜ H → ˜ H are defined byΘ ǫ,h := [ L + ǫ D h/ − M h L ǫ D − h/ ] h + L − ǫ D h/ h + M h L ǫ D − h/ . (92)If we view H ′ as a complex vector space and endow it with this inner product, we obtain a new innerproduct space whose Cauchy completion yields a Hilbert space which we denote by ˜ K . Finally, werecall that, defining (( A, A ′ )) { a } := hh U x A, U x A ′ ii ˜ η + , we can endow the space V of the solutions ofProca equation with the positive-definite inner product(( A, A ′ )) { a } := κ M (cid:2) ≺ A ( x ) , Θ + , +1 A ′ ( x ) ≻ + ≺ E ( x ) , Θ + , − E ′ ( x ) ≻ + i (cid:8) ≺ E ( x ) , Θ − , A ′ ( x ) ≻ − ≺ A ( x ) , Θ − , E ′ ( x ) ≻ (cid:9)(cid:3) , (93) This amounts to restricting the choice of the metric operator in the spirit of [34]. It does not however fix themetric operator (up to a trivial scaling) because H and Λ do not form an irreducible set of operators. The condition that H is ˜ η + -pseudo-Hermitian yields A = P ǫ = ± P h =0 , ± P h ′ =0 , ± R R d k α ǫh,h ′ ( k ) | Ψ ǫ,h ( k ) ih Φ ǫ,h ′ ( k ) | which does not commute with Λ unless α ǫh,h ′ = δ h,h ′ α ǫ,h . x and making use of (11) or equivalently (30).By construction, (93) gives the most general positive-definite inner product that renders the gen-erator of time-translations as well as the helicity operator Hermitian. We next impose the conditionthat (93) be Lorentz-invariant as well. This will give rise to the most general physically admissibleinner product on the space of Proca fields.To quantify the requirement that the right-hand side of (93) be a Lorentz scalar we compute theinner product of two plane-wave solutions of the Proca equation, i.e., (27). Using (24), (32), (87),(88), and (92) in (93), we find after a rather lengthy calculation(( A ǫ,h ( k ) , A ǫ ′ ,h ′ ( k ′ ))) { a } = κ M | N ǫ,h ( k ) | a ǫ,h ω k δ ( k − k ′ ) δ ǫ,ǫ ′ δ h,h ′ , (94)where we have introduced the abbreviation a ǫ,h := | α ǫ,h | . Because a ǫ,h are positive real numbers andthe right-hand side of (94) does not involve x , this equation provides an explicit demonstration ofthe invariance and positive-definiteness of the inner product (93).Next, we recall from Refs. [3, 46] that relativistically invariant normalization of two plane-wavesolutions is given by: (( A ǫ,h ( k ; x ) , A ǫ ′ ,h ′ ( k ′ ; x ))) = 2 ω k δ ( k − k ′ ) δ ǫ,ǫ ′ δ h,h ′ . This together with thefact that N ǫ,h ( k ) must be a Lorentz scalar so that A ǫ,h ( k ; x ) is a four-vector, shows that a ǫ,h are dimensionless positive real numbers, i.e., they do not depend on k and obey the same Lorentztransformation rule as scalars. This result ensures the relativistic invariance of the inner product ofany two solutions A and A ′ having the general form: A = X ǫ = ± X h =0 , ± Z R d k c ǫ,h ( k ) A ǫ,h ( k ) , (95)where c ǫ,h ( k ) are complex coefficients. In view of (94) and (95) and the fact that the inner product(93) is a Hermitian sesquilinear form, we obtain(( A, A ′ )) { a } = κ M X ǫ = ± X h =0 , ± Z R d k a ǫ,h ω k | N ǫ,h ( k ) | c ∗ ǫ,h ( k ) c ′ ǫ,h ( k ) . (96)Because A and A ǫ,h ( k ) are four-vectors and d k /ω k is a relativistically invariant measure [3, 46], c ǫ,h ( k ) obeys the same Lorentz transformation rule as ω − k . This in turn implies that in order for theinner product (96) to be scalars, a ǫ,h must transform as scalars. Therefore, in view of (87) and (88),we conclude that L ±± and L ± are dimensionless real numbers given by L ǫ ± = 14 [ a + , + ǫ a + , − ± a − , ± ǫ a − , − ] , L ± = 12 [ a + , ± a − , ] . (97)Inserting these relations in (92) and using (30) we findΘ + , +1 A = Θ + , D − / ˙ E , Θ + , − E = − Θ + , D − / ˙ A . These in turn allow us to express the most general physically admissible inner product on the space V of the Proca fields, namely (93), as(( A, A ′ )) { a } := κ M h ≺ A ( x ) , Θ + , D − / ˙ E ′ ( x ) ≻ − ≺ E ( x ) , Θ + , D − / ˙ A ′ ( x ) ≻− i (cid:8) ≺ A ( x ) , Θ − , E ′ ( x ) ≻ − ≺ E ( x ) , Θ − , A ′ ( x ) ≻ (cid:9)(cid:3) , (98)18here, according to Eq. (92), Θ ± , = [ L + ± − L ± ] h + L −± h + L ± . (99)Note that L ++ is a positive number and we can absorb it in the definition of the field. Therefore,the inner product (98) actually involves five nontrivial free parameters. Endowing V with this innerproduct and Cauchy completing the resulting inner product space we obtain a separable Hilbertspace which we denote by H { a } .We can express the inner product (98) in terms of the indefinite Proca inner product (41). First,we use the unitary operator U x : H { a } → ˜ K to find an expression for the action of the helicityoperator H := U − x Λ U x on a Proca field A ∈ H { a } . [ H A ]( x ) is a Proca field whose components canbe evaluated using (53), (33), and (17). This gives[ H A ]( x ) = (cid:0) , h A ( x ) (cid:1) . (100)Because [Λ , H ] = 0 and H and h are respectively related via a similarity transformation to Λ and H , we have [ H , h] = 0. This together with the identity H = H suggest to use H to split V into thesubspaces V ± and V of ±
1- and 0-helicity Proca fields according to V h := { A h ∈ V| H A h = h A h } where h ∈ {− , , +1 } . We can use H to define the ±
1- and 0-helicity components of the Proca fields A ∈ V according to: A ± := 12 ( H A ± H A ) ∈ V ± , A := ( A − H A ) ∈ V . (101)Clearly, A = A +1 + A − + A which implies V = V +1 ⊕ V − ⊕ V . Restricting the inner product(98) to V ± and V (and Cauchy completing the resulting inner product spaces) we obtain Hilbertsubspaces H { a }± and H { a } of H { a } .Next we recall that H is obtained via a unitary similarity transformation from the Hermitianoperator Λ. This in turn implies (( A, H A ′ )) { a } = (( H A, A ′ )) { a } . (102)Therefore, H is a Hermitian operator acting is H { a } . It is a physical observable that measures thehelicity of A . Furthermore, in light of (102) and (101),(( A +1 , A − )) { a } = (( A +1 , A )) { a } = (( A − , A )) { a } = 0 , ∀ A h ∈ V h , h ∈ {− , , +1 } . These relations show that H { a } = H { a } +1 ⊕ H { a }− ⊕ H { a } . Also, as we have shown in Section 3,the chirality operator C is a Hermitian involution acting in H { a } which decomposes the Hilbertspace H { a } into its ± -energy Hilbert subspaces H { a }± . Thus, using the operators H and C, we candecompose the Hilbert space into six mutually orthogonal subspaces H { a } ( ǫ,h ) , where ǫ = { + , −} , and h ∈ {− , , +1 } . Employing this decomposition of A in (98), we find(( A, A ′ )) { a } = X ǫ = ± X h =0 , ± ǫ a ǫ,h (( A ǫ,h , A ′ ǫ,h )) Σ3 . (103)19herefore, (98) is a linear combination of the indefinite inner products (41) of the components A ǫ,h of A with suitable non-negative coefficients. Note that the decomposition of the fields into the helicitycomponents (101) is clearly frame-dependent and so (103) hides the Lorentz covariance of the theory.But the inner product (98) does not involve the explicit splitting of the fields into ( ǫ, h )-components.In the remainder of this section, we obtain the Foldy transformation associated with the generalinner product (98). First, consider the invertible linear operator A of (84). This is a unitary operatormapping ˜ K onto K , for it satisfies, for all ξ, ζ ∈ ˜ K , hhA ξ, A ζ ii η + = hA ξ, η + A ζ i = h ξ, A † η + A ζ i = hh ξ, ζ ii ˜ η + . (104)We can find the explicit form of A and A − by substituting (44) and (46) in (84). This gives A = [ F + − σ M − F − σ N + ( F ++ − F ) σ ] h + [ F −− σ M + F − + σ ] h + F − σ N + F σ , (105) A − = [ ˜ F + − σ M − ˜ F − σ N + ( ˜ F ++ − ˜ F ) σ ] h + [ ˜ F −− σ M + ˜ F − + σ ] h + ˜ F − σ N + ˜ F σ , (106)where F ǫ ± := 12 (cid:2) Z + ǫ ± Z − ǫ (cid:3) , F ± := 12 [ α + , ± α − , ] , (107)˜ F ǫ ± := 12 h ˜ Z + ǫ ± ˜ Z − ǫ i , ˜ F ± := 12 (cid:2) α − , ± α − − , (cid:3) , (108) Z ǫ ± := 12 [ α ǫ, ± α ǫ, − ] , ˜ Z ǫ ± := 12 [ α − ǫ, ± α − ǫ, − ] . (109)Next, we define ˜ ρ : ˜ K → H ′ and U { a } : H { a } → H ′ by˜ ρ := ρ A , (110) U { a } = ˜ ρ U x . (111)In view of ρ † = ρ = √ η + and (85), ˜ ρ is a unitary transformation mapping ˜ K onto H ′ , h ˜ ρξ, ˜ ρζ i = h ξ, ˜ ρ † ˜ ρζ i = h ξ, A † η + A ζ i = hh ξ, ζ ii ˜ η + . (112)This in turn implies that U { a } : H { a } → H ′ is also a unitary transformation, hU { a } A, U { a } A ′ i = (( A, A ′ )) { a } , ∀ A, A ′ ∈ H { a } . (113)We can compute the explicit form of the unitary operator U { a } and its inverse. This requirescomputing ˜ ρ and its inverse. Substituting (61), (62), (105) and (106) in (110), we have˜ ρ = 12 M √ γ ˜ ρ + , + ˜ ρ − , + ˜ ρ − , − ˜ ρ + , − ! , ˜ ρ − = 12 M √ γ ˜ ρ inv+ , + − ˜ ρ inv − , − − ˜ ρ inv − , + ˜ ρ inv+ , − ! , (114)where the entries are the following operators that act in ˜ H .˜ ρ ǫ,ǫ ′ := M ( γD / + ǫD − / ) h Z ǫ ′ + h + Z ǫ ′ − h i + α ǫ ′ , ( γ M D − / + ǫD / )[ λ − h ] , (115)˜ ρ inv ǫ,ǫ ′ := M ( γD / + ǫD − / ) h ˜ Z ǫ ′ + h + ˜ Z ǫ ′ − h i + α − ǫ ′ , ( γ M D − / + ǫD / )[ λ − h ] . (116)20ext, we use (111), (114), (115), (53), and (35), to find, for all A ∈ H , U { a } A = 12 r κ M U + , + A ( x ) − i U + , − E ( x ) U − , + A ( x ) + i U − , − E ( x ) ! , (117)where the operators U ǫ,ǫ ′ : ˜ H → ˜ H , are given by U ǫ,ǫ ′ = h Z ǫ + D ǫ ′ / − M ǫ ′ α ǫ, D − ǫ ′ / i h + Z ǫ − D ǫ ′ / h + M ǫ ′ α ǫ, D − ǫ ′ / . (118)Again the arbitrary parameter γ drops out of the calculations. U − { a } = U − x ˜ ρ − is also easy tocalculate. Let ξ ∈ H ′ be a six-component vector (as in (51)) with components ξ and ξ . Then inview of (117), U − { a } ξ is the Proca field A ∈ H { a } satisfying the following initial conditions: A ( x ) = D − / √ M κ K · (cid:18) ξ α + , − ξ α − , (cid:19) , ˙ A ( x ) = − i D / √ M κ K · (cid:18) ξ α + , + ξ α − , (cid:19) , (119) A ( x ) = r M κ (cid:0) U − , + ξ + U − − , + ξ (cid:1) , ˙ A ( x ) = − i r M κ D / (cid:0) U − , + ξ − U − − , + ξ (cid:1) , (120)where U − ǫ, + is the inverse of U ǫ, + given by U − ǫ, + = (cid:20) ˜ Z ǫ + D − / − D / M α ǫ, (cid:21) h + ˜ Z ǫ − D − / h + D / M α ǫ, , (121)and we have made use of (15), (30), (33), (35), (53), and (114) – (116). By virtue of (17), for all x ∈ R , we have A ( x ) = D − / √ M κ (cid:20) e − i ( x − x ) D / K · ξ α + , − e i ( x − x ) D / K · ξ α − , (cid:21) , (122) A ( x ) = r M κ h e − i ( x − x ) D / U − , + ξ + e i ( x − x ) D / U − − , + ξ i . (123)Next we recall that since [ H, A ] = 0, we have h { a } = h. Therefore, in light of the above analysis,the pairs ( H { a } , h), ( ˜ K , H ), and ( H ′ , H ′ ) are unitarily equivalent; they represent the same quantumsystem.Finally, we give an alternative form of the Foldy transformation (117). In view of (83) and U − ǫ, − E = − U ǫ, + D − / ˙ A , we have U { a } A = r κ M U + , + A + ( x ) U − , + A − ( x ) ! . (124)In particular, setting α ǫ,h = 1, for all ǫ ∈ { + , −} and h ∈ {− , , +1 } , yields U A = r κ M U A + ( x ) A − ( x ) ! , (125)where A ± is ± -energy component of Proca field. These equations show that U ǫ, + A ǫ (specially U A ǫ )satisfy the Foldy equation [17, 16] i∂ U ǫ, + A ǫ = ǫD / U ǫ, + A ǫ . (126)21 Conserved Current Density
The relativistic and time-translation invariance of the positive-definite inner products (98) suggestthe existence of an associated conserved four-vector current density J µ { a } . In this section we use theapproach of [6] to compute J µ { a } for the case a ǫ,h = 1 and obtain a manifestly covariant expression for J µ { a } with a ǫ,h = 1 which for brevity we denote by J µ .According to (98), for all A ∈ V , (( A, A )) { a } = κ M Z R d x n A ( x , x ) ∗ · (cid:16) Θ + , D − / ˙ E ( x , x ) (cid:17) − E ( x , x ) ∗ · (cid:16) Θ + , D − / ˙ A ( x , x ) (cid:17) − i (cid:2) A ( x , x ) ∗ · (cid:0) Θ − , E ( x , x ) (cid:1) − E ( x , x ) ∗ · (cid:0) Θ − , A ( x , x ) (cid:1)(cid:3)(cid:9) . (127) In analogy with non-relativistic QM, we define the 0-component of the current density J µ { a } associatedwith A as the integrand in (127), i.e., J { a } ( x ) := κ M n A ( x ) ∗ · (cid:16) Θ + , D − / ˙ E ( x ) (cid:17) − E ( x ) ∗ · (cid:16) Θ + , D − / ˙ A ( x ) (cid:17) − i [ A ( x ) ∗ · (Θ − , E ( x )) − E ( x ) ∗ · (Θ − , A ( x ))] } . (128)Setting a ǫ,h = 1 for all ǫ ∈ { + , −} and h ∈ {− , , +1 } , yields J ( x ) := κ M n A ( x ) ∗ · D − / ˙ E ( x ) − E ( x ) ∗ · D − / ˙ A ( x ) o . (129)In order to obtain the spatial components of J µ , we follow the procedure outlined in Ref. [6]. Namely,we perform an infinitesimal Lorentz boost transformation that changes the reference frame to theone moving with a velocity v : x → x ′ = x − β · x , x → x ′ = x − β x , (130)where β := v /c . Assuming that J µ is indeed a four-vector field and neglecting the second and higherorder terms in powers of the components of β , we then find J ( x ) → J ′ ( x ′ ) = J ( x ) − β · J ( x ) . (131)Next, we use (129) to obtain J ′ ( x ′ ) := κ M n A ′ ( x ′ ) ∗ · D ′− / ˙ E ′ ( x ′ ) − E ′ ( x ′ ) ∗ · D ′− / ˙ A ′ ( x ′ ) o , (132)where x ′ := ( x ′ , x ′ ), D ′ = −∇ ′ + M , E ′ ( x ′ ) = − ˙ A ′ ( x ′ ) − ∇ ′ A ′ ( x ′ ), and ˙ A ′ ( x ′ ) := ∂ A ′ ( x ′ ) /∂x ′ . Thisreduces the determination of J to expressing the right-hand side of (132) in terms of the quantitiesassociated with the original (unprimed) frame and comparing the resulting expression with (131).Under the transformation (130), the four-vector A = ( A , A ) transforms as A ( x ) → A ′ ( x ′ ) = A ( x ) − β · A ( x ) , A ( x ) → A ′ ( x ′ ) = A ( x ) − β A ( x ) . (133)Therefore, in view of (130), we can easily obtain the transformation rule for E . The result is E ( x ) → E ′ ( x ′ ) = E ( x ) + β × ( ∇ × A ( x )) . (134)22ecause the chirality operator C = iD − / ∂ is Lorentz invariant [6], D ′− / ∂ ′ = D − / ∂ . (135)Substituting (133) – (135) in (132), making use of (131), we obtain J i ( x ) = κ M n A ∗ ( x ) · ∂ i D − / ˙ A ( x ) − [ ∂ i A ( x ) ∗ ] · D − / ˙ A ( x ) − [ A ∗ ( x ) · ∂ ] D − / ˙ A i ( x ) + [ D − / ˙ A ( x ) · ∂ ] A i ( x ) ∗ o , (136)where ∂ := ( ∂ , ∇ ) and for any two four-vectors v and v , v · v := v µ v µ . This relation suggests J µ ( x ) = κ M n A ∗ ( x ) · ∂ µ D − / ˙ A ( x ) − [ ∂ µ A ( x ) ∗ ] · D − / ˙ A ( x ) − [ A ∗ ( x ) · ∂ ] D − / ˙ A µ ( x ) + [ D − / ˙ A ( x ) · ∂ ] A µ ( x ) ∗ o . (137)It is not difficult to check (using the Proca equation) that the expression for J obtained using thisequation agrees with the one given in (129).We can use (2) and (82) to further simplify (137). This yields J µ ( x ) = iκ M { A ν ( x ) ∗ F νµc ( x ) − F νµ ( x ) ∗ A c ν ( x ) } , (138)where F µνc := ∂ µ A νc − ∂ ν A µc . The current density J µ which is generally complex-valued has thefollowing remarkable properties:1. The expression (138) for J µ is manifestly covariant; since A and A c are four-vector fields, so is J µ .2. Using the fact that both A and A c satisfy the Proca equation (4), one easily checks that J µ satisfies the following continuity equation. ∂ µ J µ = 0 . (139)Hence it is a conserved current density.Next, we use (138) to derive a manifestly covariant expression for the inner product (55) on thespace of solutions of the Proca equation (11). The result is(( A, A ′ )) = iκ M Z σ dσ ( x ) n µ ( x ) { A ν ( x ) ∗ F ′ νµc ( x ) − F νµ ( x ) ∗ A ′ c ν ( x ) } , (140)where σ is an arbitrary spacelike (Cauchy) hypersurface of the Minkowski space with volume element dσ and unit (future) timelike normal four-vector n µ . Note that in deriving (140) we have also madean implicit use of the fact that any inner product is uniquely determined by the corresponding norm[47].Using the same approach we have calculated J µ { a } for a ǫ,h = 1 and checked that indeed it is aconserved complex-valued four-vector field. But we were not able to obtain a manifestly covariantexpression for J µ { a } in this case. As the expression for J µ { a } is highly complicated we do not present ithere. 23 Physical Observables and Wave Functions for Proca Fields
The unitary equivalence of the representations ( H { a } , h), ( ˜ K , H ), and ( H ′ , H ′ ) for the quantum sys-tem describing the Proca fields allows for the construction of the observables of this system usingany of these representations. Because H ′ = ˜ H ⊕ ˜ H and ˜ H = L ( R ) ⊕ L ( R ) ⊕ L ( R ), the Foldyrepresentation ( H ′ , H ′ ) is more convenient for this purpose. In this section we construct the observ-ables in this representation and use the unitary map U { a } : H { a } → H ′ to obtain their form in thestandard (covariant) representation ( H { a } , h). Again we follow closely the approach used in [27] toconstruct the observables for KG fields.In the Foldy representation, we introduce the following set of basic observables X ′ m := x ⊗ Σ m , P ′ m := p ⊗ Σ m , S ′ m := 1 ⊗ Σ m , (141)where, x , p = ~ K , and 1 are respectively the position, momentum, and identity operators acting in L ( R ), m ∈ { , , , · · · , } , and Σ m ’s are given in Eq. (9). In the following, we will omit ‘1 ⊗ ’ forbrevity. In particular, we will identify S ′ m with Σ m .As we mentioned in Section 1, the operators X ′ , Σ , Σ , Σ , Σ , and Σ form a maximalcommuting set of observables acting in H ′ . In particular, we can use their common eigenvectors,namely ξ ( ǫ,s ) x := | x i ⊗ e ǫ,s , x ∈ R , ǫ ∈ {− , + } , s ∈ {− , , +1 } , (142)to construct a basis of H ′ . In (142), e ǫ,s are the vectors defined in (10) and | x i are the δ -functionnormalized position kets satisfying x | x i = x | x i , h x | x ′ i = δ ( x − x ′ ), and R R d x | x ih x | = 1. It iseasy to see that indeed X ′ ξ ( ǫ,s ) x = x ξ ( ǫ,s ) x , Σ ξ ( ǫ,s ) x = ǫ ξ ( ǫ,s ) x , and Σ ξ ( ǫ,s ) x = s ξ ( ǫ,s ) x . Furthermore, h ξ ( ǫ,s ) x , ξ ( ǫ ′ ,s ′ ) x ′ i = δ ǫ,ǫ ′ δ s,s ′ δ ( x − x ′ ) , X ǫ = ± X s =0 , ± Z R d x | ξ ( ǫ,s ) x ih ξ ( ǫ,s ) x | = Σ . (143)We can express any six-component vector Ψ ′ ∈ H ′ in the basis { ξ ( ǫ,s ) x } according toΨ ′ = X ǫ = ± X s =0 , ± Z R d x f ( ǫ, s, x ) ξ ( ǫ,s ) x , (144)where f : {− , + } × {− , , +1 } × R → C is the wave function associated with Ψ ′ in the position-representation, i.e., f ( ǫ, s, x ) := h ξ ( ǫ,s ) x , Ψ ′ i . (145)Also, the action of a physical observable O ′ : H ′ → H ′ on the state vector Ψ ′ ∈ H ′ is given by O ′ Ψ ′ = X ǫ = ± X s =0 , ± Z R d x [ ˆ Of ( ǫ, s, x )] ξ ( ǫ,s ) x , (146)where ˆ Of ( ǫ, s, x ) := X ǫ ′ = ± X s ′ =0 , ± Z R d x ′ f ( ǫ ′ , s ′ , x ′ ) h ξ ( ǫ,s ) x , O ′ ξ ( ǫ ′ ,s ′ ) x ′ i . (147)24his provides the representation of observables in terms of linear operators acting on the square-integrable wave functions f .Next, we introduce the operators x m { a } := U − { a } X ′ m U { a } , p m { a } := U − { a } P ′ m U { a } , s m { a } := U − { a } Σ m U { a } , (148)that act in H { a } , and define the Proca fields A ( ǫ,s ) { a } x = ( A ǫ,s ) { a } x , A ( ǫ,s ) { a } x ) := U − { a } ξ ( ǫ,s ) x , (149)which form a complete orthonormal basis of H { a } :(( A ( ǫ,s ) { a } x , A ( ǫ ′ ,s ′ ) { a } x ′ )) { a } = δ ǫ,ǫ ′ δ s,s ′ δ ( x − x ′ ) , X ǫ = ± X s =0 , ± Z R d x | A ( ǫ,s ) { a } x )( A ( ǫ,s ) { a } x | = s { a } . (150)Here we have used (113), (143), and (149), s { a } coincides with the identity operator for H { a } , andfor all A, A ′ ∈ H { a } , the operator | A )( A ′ | is defined by | A )( A ′ | A ′′ := (( A ′ , A ′′ )) { a } A , for any A ′′ ∈ H { a } .In view of (150) and(( A ( ǫ,s ) { a } x , A )) { a } = hU { a } A ( ǫ,s ) { a } x , U { a } A i = h ξ ( ǫ,s ) x , Ψ ′ i = f ( ǫ, s, x ) , (151)we can express any Proca field A ∈ H { a } in the basis { A ( ǫ,s ) { a } x } according to A = X ǫ = ± X s =0 , ± Z R d x f ( ǫ, s, x ) A ( ǫ,s ) { a } x . (152)Here we do not label the wave functions f with the subscript { a } , because they do not depend onthe choice of the parameter a ǫ,h . The proof of this assertion uses the unitary operator U { a } := U − { a } U , (153)that maps H onto H { a } and is identical with the one given in [6] for the KG fields.The physical observables o { a } : H { a } → H { a } are uniquely specified in terms of their representationin the basis { A ( ǫ,s ) { a } x } ; for all A ∈ H { a } o { a } A = X ǫ = ± X s =0 , ± Z R d x [ ˆ Of ( ǫ, s, x )] A ( ǫ,s ) { a } x , (154)where ˆ Of ( ǫ, s, x ) is defined by (147). This follows from (( A ( ǫ,s ) { a } x , o { a } A ( ǫ ′ ,s ′ ) { a } x ′ )) { a } = h ξ ( ǫ,s ) x , O ′ ξ ( ǫ ′ ,s ′ ) x ′ i .In view of (150) and (152), we can express the transition amplitudes between two states (the innerproduct of two Proca fields) in the form(( A, A ′ )) { a } = X ǫ = ± X s =0 , ± Z R d x f ( ǫ, s, x ) ∗ f ′ ( ǫ, s, x ) . (155)More generally, for any observable o { a } : H { a } → H { a } , we have(( A, o { a } A ′ )) { a } = X ǫ = ± X s =0 , ± Z R d x f ( ǫ, s, x ) ∗ ˆ Of ′ ( ǫ, s, x )= X ǫ = ± X s =0 , ± Z R d x [ ˆ Of ( ǫ, s, x )] ∗ f ′ ( ǫ, s, x ) . (156)25he above discussion shows that we may view the wave functions f as elements of H ′ , and similarlyto the case of KG fields [26, 27], formulate the QM of Proca fields in terms of these wave functions.In this formulation the observables are Hermitian operators ˆ O acting on the wave functions. Forexample, the action of x { a } , p { a } , s { a } , and s { a } on A corresponds to the action of the operatorsˆ x { a } , ˆ p { a } , ˆ s { a } , and ˆ s { a } on f , whereˆ x { a } f ( ǫ, s, x ) := x f ( ǫ, s, x ) , ˆ p { a } f ( ǫ, s, x ) := − i ~ ∇ f ( ǫ, s, x ) , (157)ˆ s { a } f ( ǫ, s, x ) := ǫf ( ǫ, s, x ) , ˆ s { a } f ( ǫ, s, x ) := sf ( ǫ, s, x ) . (158)These equations show that f ( ǫ, s, x ) are the position wave functions with definite chirality (sign ofthe energy) ǫ and spin s (say along the x -direction). They are however not the eigenfunctions of thehelicity operator. To see this, we recall from (40) that H ′ = ˜ H ⊕ ˜ H , and the spin operator acting in˜ H is given by S of Eq. (25). This shows that the spin operator acting in H ′ is S ′ := (Σ , − Σ , Σ ) = S S ! . (159)Denoting the spin operator acting in H { a } by s { a } , we have s { a } = U − { a } S ′ U { a } . (160)The projection of this operator along the momentum p { a } gives the helicity operator:(ˆ p { a } · ˆ s { a } ) f ( ǫ, s i , x ) = ε ijk ∂ j f ( ǫ, s k , x ) , (161)where ( s , s , s ) = (+1 , − , A corresponds to the action of the operator ˆh :=ˆ s { a } q ˆ p { a } + m c on the wave function f :ˆh f ( ǫ, s, x ) = ~ ǫ √−∇ + M f ( ǫ, s, x ) . (162)Subsequently, in view of (162), and (58), the dynamics of the evolving Proca field A x is determinedin terms of the wave functions f ( ǫ, s, x ; x ) = (( A ( ǫ,s ) { a } x , A x )) { a } , according to i ~ ∂ f ( ǫ, s, x ; x ) = ǫ √− ~ ∇ + m c f ( ǫ, s, x ; x ) . (163)Furthermore, applying i∂ to both sides of (163), we can check that the wave functions f also satisfiesthe KG equation: [ ∂ − ∇ + M ] f ( ǫ, s, x ; x ) = 0.Next, recall that because the time-reversal operator (79) acting in H ′ commutes with X ′ , theeigenvectors ξ ( ǫ,s ) x may be taken to be real. In this case the action of the time-reversal operator T = PT on any A ∈ H { a } is equivalent to the complex-conjugation of the associated wave-function f , i.e., ˆ T f ( ǫ, s, x ) = f ( ǫ, s, x ) ∗ . Similarly, we can identify the operators ˆ s and ˆ s , respectively, withthe chirality and spin operators acting on the wave functions f .As we shall see in the following section the wave functions f ( ǫ, s, x ) furnish a position represen-tation for the QM of the Proca fields. The corresponding position operator is the spin-1 analog ofthe Newton-Wigner position operator for the KG field [21, 22] and similarly to the latter fails to berelativistically covariant. This means that the above-mentioned position-representation provides Throughout this paper we express spin and angular momentum operators in units of ~ . See however [48].
26 non-covariant description of a quantum system that also admits a unitary-equivalent covariantdescription in terms of the Hilbert space H { a } and the Hamiltonian h. As discussed in [27] for the KG fields, the canonical quantization scheme that provides the physicalmeaning of the observables yields the Foldy representation of the quantum system. This suggeststhat the operators ˆ x { a } and ˆ p { a } that clearly satisfy the canonical commutation relations may beidentified with the position and momentum operators acting on the wave functions f . This in turnmeans that the operators X ′ and P ′ in the Foldy representation and the operators x { a } and p { a } in the ( H { a } , h)-representation also describe the position and momentum observables. In particular,the basis vectors ξ ( ǫ,s ) x and A ( ǫ,s ) { a } x determine the states of the system with a definite position value x ;they are localized in space. They also have definite charge or chirality (sign of the energy) and spin(say along the x -direction).By construction, A ( ǫ,s ) { a } x are delta-function normalized position eigenvectors, i.e., they satisfy (150)and x { a } A ( ǫ,s ) { a } x = x A ( ǫ,s ) { a } x . (164)Similarly, we identify the chirality operator and the spin operator along the x -direction with C = U − { a } Σ U { a } and s = U − { a } Σ U { a } , respectively. This impliesC A ( ǫ,s ) { a } x = ǫ A ( ǫ,s ) { a } x , s A ( ǫ,s ) { a } x = s A ( ǫ,s ) { a } x . (165)In view of Eqs. (150), (164), and (165), the state vectors A ( ǫ,s ) { a } x represent spatially localized Procafields with definite chirality ǫ and spin s . We can associate each Proca field A ∈ H { a } with a uniqueposition wave function, namely f ( ǫ, s, x ). As we explained in Section 6, we can use these wavefunctions to represent all the physical quantities associated with the Proca fields. We also emphasizethat according to (161), f ( ǫ, s, x ) are not the helicity eigenfunctions. Therefore, the state vectors A ( ǫ,s ) { a } x do not have definite helicity. This is in complete accordance with the Heisenberg uncertaintyprinciple: since A ( ǫ,s ) { a } x are localized states (with definite position) they do not have definite momentumand hence definite helicity. Next, we recall that two Hilbert spaces H { a } for different choices of the parameters { a } are unitary-equivalent. We can use (148), (149), and the unitary operator (117) to obtain the explicit form ofthe localized Proca fields and the physical observables acting in H { a } . The resulting expressions forthe position and spin operators are highly complicated. Therefore, in the following we shall onlyderive the explicit form of these operators in the covariant representation ( H , h). In the classical limit, each component of a Proca field corresponds to a classical free particle of energy E = ± p p + m c . Upon quantization E → ± ~ √−∇ + M which signifies the relevance of the Foldy representation. .1 Explicit Form of the Localized States As suggested by (155), the wave functions f ( ǫ, s, x ) belong to L ( R ). Moreover due to the implicitdependence of A ( ǫ,s ) x on the x appearing in the expression for U , f ( ǫ, s, x ) depend on x . As in thecase of KG fields this dependence becomes explicit, if we express f ( ǫ, s, x ) in terms of A directly. Tosee this, we first substitute (67), (68) and (142) in (149) to obtain A ǫ,s i ) x ( x ) = ǫ D − / √ M κ e − iǫ ( x − x ) D / K i | x i , A ( ǫ,s i ) x ( x ) = r M κ U − e − iǫ ( x − x ) D / | x i ⊗ e s i , (166)where ( s , s , s ) = (+1 , − , f ( ǫ, s i , x ) = r κ M e iǫ ( x − x ) ˆ D / (cid:2) U A ǫ ( x , x ) (cid:3) i , (167)where A µǫ is the definite-chirality (definite-energy) component of A µ with chirality ǫ . Because A µǫ andconsequently U A ǫ satisfy the Foldy equation (126), we have e iǫ ( x − x ) D / U A ǫ ( x , x ) = U A ǫ ( x , x ).Inserting this in (167), we find the following manifestly x -dependent expression for f ( ǫ, s i , x ). f ( ǫ, s i , x ) = r κ M (cid:2) U A ǫ ( x , x ) (cid:3) i . (168)Next, we use (166) to compute the value of the localized Proca fields A ( ǫ,s i ) y at a spacetime point( x , x ): A ( ǫ,s i ) y ( x , x ) := h x | A ( ǫ,s i ) y ( x ) i . (169)Doing the necessary calculations, we obtain A ǫ, +1) y ( x ) = iǫ sin θ cos φ I , A ǫ, − y ( x ) = iǫ sin θ sin φ I , A ǫ, y ( x ) = iǫ cos θ I , (170) A ( ǫ, +1) y ( x ) = v v v , A ( ǫ, − y ( x ) = v v v , A ( ǫ, y ( x ) = v v v , (171) where x := ( x , x ), θ and φ are the polar and azimuthal angles representing the direction of x − y , v := I + sin θ cos φ I , v := sin θ sin(2 φ ) I , v := sin(2 θ ) cos φ I ,v := I + sin θ sin φ I , v := sin(2 θ ) sin φ I , v := I + cos θ I , and I := | x − y | π √ M κ Z ∞ dk k Ω ( ǫ, k ) Ω ( k ) , (172) I := r M κ π Z ∞ dk Ω ( ǫ, k ) (cid:26) k sin( k | x − y | ) | x − y | + Ω ( k ) (cid:8) M− [ k + M ] / − (cid:9)(cid:27) , (173) I := r M κ π Z ∞ dk Ω ( ǫ, k ) (cid:26) k sin( k | x − y | ) | x − y | − ( k ) (cid:27) (cid:8) M− [ k + M ] / − (cid:9) , (174)Ω ( ǫ, k ) := exp (cid:2) − iǫ ( x − x ) √ k + M (cid:3) [ k + M ] / , Ω ( k ) := sin( k | x − y | ) k | x − y | − cos( k | x − y | ) | x − y | . (175)28 .1 0.15 0.2 0.25 0.3 0.350246810 I j ( M √ κ × ) | x − y | ( M − × − ) I I I Figure 1: Plots of I , I and I in terms of the radial distance | x − y | . The radial distance is scaledwith the Compton wave length M− .For x = x , the integrals on the right-hand side (172) – (174) can be expressed in terms of the BesselK-function ( K n ), and the Hypergeometric function ( p F q ). The result, for both ǫ = − I = r M κ h π Γ( ) i − (cid:16) M z (cid:17) (cid:26) M z K ( M z ) + K ( M z ) (cid:27) , (176) I = r M κ h π Γ( ) i − (cid:16) M z (cid:17) ( K ( M z ) + 1 M z K ( M z ) + Γ( ) π ( M z ) K ( M z )+1( M z ) (cid:18) π Γ( ) F (cid:20)
12 ; 54 ,
32 ; M z (cid:21) + Γ( ) π. F (cid:20)
12 ; 32 ,
74 ; M z (cid:21)(cid:19) +Γ( )( M z ) (cid:18) M z F (cid:20)
14 ; 14 ,
34 ; M z (cid:21) − F (cid:20)
14 ; 34 ,
54 ; M z (cid:21)(cid:19)) , (177) I = r M κ h π Γ( ) i − (cid:16) M z (cid:17) ( K ( M z ) M z + K ( M z )( M z ) + Γ( ) π ( M z ) " K ( M z ) + 3 K ( M z ) M z +1(2 M z ) (cid:18) π Γ( ) F (cid:20)
12 ; 54 ,
32 ; M z (cid:21) + Γ( ) π F (cid:20)
12 ; 32 ,
74 ; M z (cid:21)(cid:19) +3Γ( )( M z ) (cid:18) M z F (cid:20) −
14 ; 14 ,
34 ; M z (cid:21) + 2 F (cid:20)
14 ; 34 ,
54 ; M z (cid:21)(cid:19)) , (178) where z := | x − y | and Γ is the Gamma function. Fig. 1 gives the graphs of I , I , and I . Theyinvolve a δ -function-like singularity at | x − y | = 0. This is a manifestation of the fact that A ( ǫ,s ) y arelocalized at y .As seen from Eqs. (170) and (171), the localized Proca fields, unlike the localized KG fields [6],depend on the angles θ and φ (direction of x − y ). But as for the KG fields their position wavefunctions involve the delta functions; the position wave function f ( ǫ,s, x ) ( ǫ ′ , s ′ , x ′ ) for A ( ǫ,s ) x ( x ) has theform δ ǫ,ǫ ′ δ s,s ′ δ ( x − x ′ ).Finally, we wish to emphasize that to the best of our knowledge the explicit form of the localizedProca fields A ( ǫ,s ) x have not been previously given. It is remarkable that we have obtained these29ocalized states without pursuing the axiomatic approach of Ref. [21]. The latter gives rise to theBargmann-Wigner localized fields [21, 20]. Using the unitary map U and its inverse, we can obtain the explicit form of the position operator x that is defined to act on the Proca fields A ∈ H . Note that χ := x A is a three-component fieldwhose components satisfy both (11) and (13). Therefore it is uniquely determined in terms of theinitial data ( χ ( x ) , ˙ χ ( x )) for some x ∈ R . One can compute the latter using (148), (65), (67), (68),and the identities: D = K + M , h S h = K | K | h , (179)[ x , h ] = i | K | S − i K | K | h , (cid:2) x , h (cid:3) = i | K | ( h S + S h ) − i K | K | h , (180)and [ F ( K ) , x ] = − i ∇ K F ( K ), where F is a differentiable function. This yields χ ( x ) = Y A ( x ) + 1 M D / ˙ A ( x ) , ˙ χ ( x ) = (cid:18) Y − i K D (cid:19) ˙ A ( x ) − D / M A ( x ) , (181) χ ( x ) = X A ( x ) , ˙ χ ( x ) = (cid:18) X − i K D (cid:19) ˙ A ( x ) , (182) where Y := x + i K D + i K M ( D / + M ) , (183) X := x − i K D − + i K [ D − − M− D − / ] h + i | K | [1 − M D − / ] h S + i | K | [ M− D / − S h , = x − i K D − i K ( K · S ) M D ( D / + M ) + i [ S ( K · S ) + ( K · S ) S ]2 M D / − ( D / − M ) ( K × S )2 M D / ( D / + M ) . (184)Next, we employ (17) to express χ ( x ) in terms of (181) and (182). Simplify the resultingexpression, we find, for all x ∈ R , χ ( x ) = (cid:8) Y − i ( x − x ) K D − ∂ (cid:9) A ( x ) + 1 M D / ˙ A ( x ) , (185) χ ( x ) = (cid:8) X − i ( x − x ) K D − ∂ (cid:9) A ( x ) . (186)In addition to being a Hermitian operator acting in the physical Hilbert space H , the positionoperator x has the following notable properties.1. It respects the charge superselection rule [49], for it commutes with the chirality operatorC = s . This is easily seen by noting that x and s are respectively obtained via a similaritytransformation (148) from X ′ and S ′ = Σ , and that according to (141), [ X ′ , Σ ] = 0.2. It has commuting components, and it commutes with the spin operator. This is because it isobtained via a similarity transformation (148) from X ′ which has commuting components andcommutes with S ′ . 30. Its x -derivative gives the relativistic velocity operator; i.e., we have: v := d x dx = i [h , x ] = i U − [ H ′ , X ′ ] U = p p p + m c C . As we noticed in [28], this result means that the physical momentum is p C.4. It has the correct nonrelativistic limit: As c → ∞ , x → x .Similarly, we can obtain the explicit form of the spin operator s acting on A ∈ H . Again, notingthat S := s A is a three-component field whose components satisfy (11) and (13), we can determine S in terms of the initial data ( S ( x ) , ˙ S ( x )). Using (65), (67), (68), (148), (179), and[ S , h ] = i | K | ( K × S ) , (cid:2) S , h (cid:3) = i | K | [( K × S ) h + h ( K × S )] , (187)these initial values are given by S ( x ) = M− D − / K × ˙ A ( x ) , ˙ S ( x ) = − M− D / K × A ( x ) ,~ S ( x ) = ˜ s A ( x ) , ˙ ~ S ( x ) = ˜ s ˙ A ( x ) , (188)where ˜ s = S + i | K | [1 − M D − / ] h ( K × S ) + i | K | [ M− D / − K × S ) h = D + M M D / S − ( D / − M ) K ( K · S )2 M D / ( D / + M ) + i { ( K · S )( K × S ) + ( K × S )( K · S ) } M D / . (189)Next, we use (17) to express S ( x ) in terms of the initial data (188). This leads to S ( x ) = M− D − / K × ˙ A ( x ) , ~ S ( x ) = ˜ s A ( x ) . (190)As seen from (185), (186), and (190) the action of the position and spin operators on a Proca field A mixes the components of the field. It is clearly more complicated than the action of the correspondingoperators for the KG fields [6].Next, we evaluate the action of the momentum, angular momentum, and helicity operators on A . Because P ′ and ρ commute, in view of (148) and (59), we have p = U − x P ′ U x . This in turnimplies [ p A ]( x ) = p A ( x ) for all x ∈ R . Furthermore, using (185), (186), and (190), we can showthat the angular momentum operator (in units of ~ ) L := ~ − x × p acts on A according to L ( x ) = ~ − ( x × p ) A ( x ) − S ( x ) ,~ L ( x ) = { ~ − x × p + S } A ( x ) − ~ S ( x ) , ∀ x ∈ R , (191)where L ( x ) := [ L A ]( x ), and S ( x ) := [ s A ]( x ) is given in (190). Therefore, unlike the position x and spin s operators, the (linear) momentum p and the total angular momentum M := L + s operators have the same expressions as in nonrelativistic QM. Similarly, we can compute the action Note that the corresponding particle in nonrelativistic QM is described by A [46], i.e., we should set A = 0.
31f the helicity operator on A . In view of (190), we find: [( p · s ) A ]( x ) = (cid:0) , ( p · S ) A ( x ) (cid:1) , whichis in complete accordance with our previous result (100).Again, to the best of our knowledge, the actions of the position, spin, and other observables on a(covariant) Proca field A have not been previously given. Earlier works on the subject [19, 20, 21, 22]calculate the position and spin operators that act either on the six-component fields, i.e., in theHilbert space K , or on the Bargmann-Wigner’s second-rank 4-spinors. One can use the unitarytransformation ρ : K → H ′ to compute the position ( X := ρ − X ′ ρ ) and spin ( S := ρ − S ′ ρ )operators acting in K . The result is X = x − D / − M M D / ( D / + M ) ( K × S ) + σ (cid:26) − i K D − i K ( K · S ) M D ( D / + M ) + i [ S ( K · S ) + ( K · S ) S ]2 M D / (cid:27) , (192) S = D + M M D / S − D / − M M D / ( D / + M ) K ( K · S ) + i M D / σ { ( K · S )( K × S ) + ( K × S )( K · S ) } . (193) These are exactly the position and spin operators that are obtained by Case in [16].Following the treatment of the KG fields in [27, 28] we can identify the coherent states of Procafields with the eigenstates of the annihilation operator a := q k ~ ( x + ik − p ), where k ∈ R .Because both x and p commute with the chirality ( s ) and spin ( s ) operators, so does a . Hence,we can introduce a set of coherent states with definite chirality and spin. The corresponding statevectors | z , ǫ, s ) are defined as the common eigenvectors of a , s and s , i.e., a | z , ǫ, s ) = z | z , ǫ, s )and s | z , ǫ, s ) = ǫ | z , ǫ, s ), s | z , ǫ, s ) = s | z , ǫ, s ), where z ∈ C , ǫ ∈ {− , + } and s ∈ {− , , +1 } . Wecan studied these coherent states and found that they have essentially the same properties as thecoherent states of the KG fields that we explored in [28]. We may employ the procedure outlined in [6] to find the probability density for the spatial localizationof a Proca field. As in nonrelativistic QM, we identify the probability of the localization of a Procafield A in a region V ⊆ R , at time t = x /c , with P V = Z V d x k Π x A k , (194)where Π x is the projection operator onto the eigenspace of x with eigenvalue x , i.e.,Π x = X ǫ = ± X s =0 , ± | A ( ǫ,s ) x )( A ( ǫ,s ) x | , (195) k · k := (( · , · )) is the square of the norm of H , and we assume k A k = 1. Substituting (195) in (194)and making use of (150) and (151), we have P V = P ǫ = ± P s =0 , ± R V d x | f ( ǫ, s, x ) | . Therefore, inlight of (168), (82), and (83), the probability density is given by ̺ ( x , x ) := X ǫ = ± X s =0 , ± | f ( ǫ, s, x ) | = κ M n | U A ( x , x ) | + | U D − / ˙ A ( x , x ) | o . (196)For a position measurement to be made at time t = x /c , we have the probability density ̺ ( x , x ) = κ M n | U A ( x , x ) | + | U D − / ˙ A ( x , x ) | o = κ M (cid:8) | U A ( x ) | + | U A c ( x ) | (cid:9) . (197)32e can use the method discussed in Section 5 to introduce a current density J µ such that J = ̺ (See also [50].) This probability current density turns out to have the form J µ ( x ) = κ M ℜ n U A ( x ) ∗ · (cid:16) D − ∂ µ U ˙ A ( x ) (cid:17) + U A c ( x ) ∗ · (cid:16) D − ∂ µ U ˙ A c ( x ) (cid:17)o + Υ µ ( x ) , (198)where ℜ means “the real part of”, Υ ( x ) = 0 and, for all i ∈ { , , } ,Υ i ( x ) = κ M ℜ ( ( U A ( x )) i ∗ (cid:20) ∇ · D − / D / + M U ˙ A ( x ) (cid:21) − (cid:2) U A ( x ) i ∗ · ∇ (cid:3) (cid:18) D − / D / + M U ˙ A ( x ) (cid:19) i +( U A c ( x )) i ∗ (cid:20) ∇ · D − / D / + M U ˙ A c ( x ) (cid:21) − (cid:2) U A c ( x ) i ∗ · ∇ (cid:3) (cid:18) D − / D / + M U ˙ A c ( x ) (cid:19) i ) . (199)One can easily show that J µ ( x ) is neither a four-vector nor a conserved current density.Although the above discussion is based on a particular choice for the parameters { a } , namely { a } = { } , it is generally valid. This is because the position wave functions f and the correspondingprobability densities do not depend on the parameters { a } . Therefore, if we are to compute theprobability density ̺ { a } of the spatial localization of a Proca field A ∈ H { a } with the position operatorbeing identified with x { a } for { a } 6 = { } , we have, for a measurement made at t = x /c , ̺ { a } ( x , x ) = κ M n | U A ′{ a } ( x , x ) | + | U D − / ˙ A ′{ a } ( x , x ) | o , (200)where A ′{ a } := U − { a } A and U { a } : H → H { a } is given by (153). We can compute A ′{ a } using (65),(120), and (153) and use the result to obtain ̺ { a } ( x ) = κ M (cid:8) | U + , + A ( x ) | + | U − , + A ( x ) | + | U + , + A c ( x ) | + | U − , + A c ( x ) | +2 ℜ [( U + , + A ( x )) ∗ · U + , + A c ( x ) − ( U − , + A ( x )) ∗ · U − , + A c ( x )] } . (201)Again we can use the method of Section 5 to compute a probability current density J µ { a } such that J { a } = ̺ { a } . This leads to a complicated expression that we do not include here. Similarly to J µ ,we expect J µ { a } to be neither covariant nor conserved.The non-conservation (respectively non-covariance) of the probability current density J µ { a } raisesthe issue of the non-conservation (respectively frame-dependence) of the total probability: P { a } := Z R d x ̺ { a } ( x , x ) . (202)This would certainly be unacceptable. The situation is analogous to that of the KG fields. P { a } is indeed a frame-independent conserved quantity, thanks to the covariance and conservation of thecurrent density J µ { a } and the identity Z R d x ̺ { a } ( x , x ) = Z R d x J { a } ( x , x ) , (203)which follows from (128), (201) and the fact that U ǫ,ǫ ′ are self-adjoint operators acting in ˜ H .33ombining (202) and (203), we have P { a } = Z R d x J { a } ( x , x ) . (204)This relation implies that although the probability density ̺ { a } is not the zero-component of a con-served four-vector current density, its integral over the whole space that yields the total probability(202) is nevertheless conserved. Furthermore, this global conservation law stems from a local con-servation law, i.e., a continuity equation for a four-vector current density namely J µ { a } . In this section we explore a global gauge symmetry that supports the conservation of the totalprobability or its local realization as the conservation of the current density J µ { a } . To determine thenature of this symmetry, we recall that the conserved charge associated with any conserved current isthe generator of the corresponding infinitesimal gauge transformations [51]. We use the Hamiltonianformulation to obtain these transformations. The procedure we follow mimics the one presented in[6] for the KG fields.The Lagrangian L for a Proca field A and the corresponding canonical momenta Π( x ), ¯Π( x )associated with A ( x ) := A ( x , x ) and A ∗ ( x , x ) are respectively given by [3]: L := − Z R d x (cid:26) F µν ( x ) ∗ F µν ( x ) + M A µ ( x ) ∗ A µ ( x ) (cid:27) , (205)Π ( x ) := δLδ ˙ A ( x ) = 0 , Π i ( x ) := δLδ ˙ A i ( x ) = − F i ( x ) ∗ = − E i ( x ) ∗ , (206)¯Π ( x ) := δLδ ˙ A ( x ) ∗ = 0 , ¯Π i ( x ) := δLδ ˙ A i ( x ) ∗ = − F i ( x ) = − E i ( x ) . (207)The fact that Π and ¯Π vanish show that the Proca system is a constrained system. There are twoprimary constraints: Φ := Π ( x ) ≈ , Φ := ¯Π ( x ) ≈ . (208)Solving for the velocities ˙ A i and ˙ A ∗ i in (206) and (207), and using the Hamiltonian H := Z R d x (cid:8) Π i ( x ) ¯Π i ( x ) − Π i ( x ) ∂ i A ( x ) − ¯Π i ( x ) ∂ i A ( x ) ∗ + M A ( x ) A ( x ) ∗ + M A i ( x ) A i ( x ) ∗ + ∂ i A j ( x ) ∂ i A j ( x ) ∗ − ∂ i A j ( x ) ∂ j A i ( x ) ∗ (cid:9) , (209)we obtain the so-called total Hamiltonian [52]: H T := H + u j Φ j , where u j , j = 1 , {F , G} P := Z R d x (cid:20) δ F δA µ ( x ) δ G δ Π µ ( x ) − δ G δA µ ( x ) δ F δ Π µ ( x ) + δ F δA µ ( x ) ∗ δ G δ ¯Π µ ( x ) − δ G δA µ ( x ) ∗ δ F δ ¯Π µ ( x ) (cid:21) , (210) Throughout this section we suppress the x -dependence of the fields for simplicity. Following Dirac’s notation [52], we write the constraints as weak equations with the weak equality symbol ‘ ≈ ’.
34f the observables F and G , we can easily show that the dynamical consistency of the primaryconstraints (208), i.e., ˙Φ j = { Φ j , H T } P ≈
0, results in the following secondary constraints,Φ := M A ( x ) + ∂ i ¯Π i ( x ) ≈ , Φ := M A ( x ) ∗ + ∂ i Π i ( x ) ≈ , (211)and that there is no other secondary constraint.It is not difficult to show that the matrix C jj ′ ( y , z ) := { Φ j ( y ) , Φ j ′ ( z ) } P is nonsingular. Hence wehave a theory with four second-class constraints and can apply Dirac’s canonical quantization thatuses the Dirac bracket [52]: {F , G} D := {F , G} P − Z R d y Z R d z {F , Φ j ( y ) } P C jj ′ ( y , z ) { Φ j ′ ( z ) , G} P , (212)where C jj ′ ( y , z ) is the inverse of C jj ′ ( y , z ). Computing the latter and its inverse, we find C jj ′ ( y , z ) :=˜ C jj ′ δ ( y − z ), and {F , G} D = {F , G} P − Z R d y {F , Φ j ( y ) } P ˜ C jj ′ { Φ j ′ ( y ) , G} P , (213)where ˜ C = ˜ C = − ˜ C = − ˜ C = M− , and ˜ C jj ′ = 0 for other j ’s and j ′ ’s. Further details of theconstraint quantization of Proca system can be found in [53, 54, 55].In terms of the canonical phase space variables ( A, Π) and ( A ∗ , ¯Π), the total probability (204)takes the form P { a } = κ M Z R d x (cid:26) Π i ( x ) h Θ + , D − / ˙ A ( x ) i i − A i ( x ) ∗ h Θ + , D − / ˙¯ Π ( x ) i i + i (cid:16) Π i ( x ) [Θ + , A ( x )] i − A i ( x ) ∗ (cid:2) Θ + , ¯ Π ( x ) (cid:3) i (cid:17)o , (214)where we have made use of (128), (206), and (207). Now, we can obtain the infinitesimal symmetrytransformation, A → A + δA, (215)generated by P { a } using δA ( x ) = (cid:8) A ( x ) , P { a } (cid:9) D δφ, (216)where δφ is an infinitesimal real parameter. In view of (206), (207), (210), (213) – (216), (29), (33),and (82), we have δA ( x ) = − iδθ (cid:8) L C + L − (cid:9) A ( x ) , (217) δ A ( x ) = − iδθ { Θ + , C + Θ − , } A ( x ) , (218)where δθ := κ δφ/ (2 M ) and Θ ± , are given by (99). We may employ (100) to express (217) and (218)as δA ( x ) = − iδθ { ϑ + , C + ϑ − , } A ( x ) , (219)where ϑ ǫ, : H { a } → H { a } is defined by ϑ ǫ, := [ L + ǫ − L ǫ ] H + L − ǫ H + L ǫ . (220)35n view of (219), the symmetry transformations (215) are generated by the operator ϑ + , C + ϑ − , . We can easily exponentiate the latter to obtain the corresponding non-infinitesimal symmetrytransformations, A → e − iθ ( ϑ + , C+ ϑ − , ) A, (221)where θ ∈ R is arbitrary. In terms of the ( ǫ, h )-components A ǫ,h of A , (221) takes the form A = X ǫ = ± X h =0 , ± A ǫ,h → X ǫ = ± X h =0 , ± e − iǫ a ǫ,h A ǫ,h , (222)where we have made use of C = 1 and H = H .Similarly to its spin-zero counterpart [6], as seen from (221) and (222), the gauge group G { a } associated with these transformations is a one-dimensional connected Abelian Lie group. Therefore,it is isomorphic to either of U (1) or R + , the latter being the noncompact multiplicative group ofpositive real numbers [56].We can construct a faithful representation of the group G { a } using the six-component represen-tation A = ( A + , +1 , A + , − , A + , , A − , +1 , A − , − , A − , ) T where C and H are, respectively, represented byΣ and Σ . In this representation a typical element of G { a } takes the form g a ( θ ) := diag (cid:0) e − i a + , +1 θ , e − i a + , − θ , e − i a + , θ , e i a − , +1 θ , e i a − , − θ , e i a − , θ (cid:1) . (223)This expression suggests that the gauge group G { a } is a subgroup of U (1) ⊗ , the latter being thedirect product of six copies of U (1). It is not difficult to show that G { a } is a compact subgroup ofthis group and consequently isomorphic to U (1) if and only if all the parameters a ǫ,h are rationalnumbers, otherwise G { a } is isomorphic to R + .Clearly, the G { a } gauge symmetry associated with the conservation of the total probability is aglobal gauge symmetry. Similarly to its spin-zero counterpart [6] the local analog of this global gaugesymmetry is different from the usual local Yang-Mills-type gauge symmetries.
10 Conclusion and Discussion
In this article we have used the methods of pseudo-Hermitian quantum mechanics to devise a com-plete formulation of the relativistic quantum mechanics of the Proca fields that does not involverestricting to the positive-energy solutions of the Proca equation. In particular, we have constructedthe most general physically admissible inner product (( · , · )) { a } on the solution space of the Procaequation. Up to a trivial scaling, this inner product involves five real parameters that we collectivelydenote by a . For all the values of a the inner product (( · , · )) { a } is positive-definite and relativisti-cally invariant. It also renders the generator of the time-translations, i.e., the Hamiltonian h, andthe helicity operator self-adjoint. The quantum system associated with the Proca fields may berepresented by a Hilbert space H { a } defined by the inner product (( · , · )) { a } and the Hamiltonian h. Here we identify the gauge group with its connected component that includes the identity and is obtained byexponentiating the generator ϑ + , C + ϑ − , . H { a } , and the localized states ofthe Proca field. Furthermore, we introduced the position wave functions and used them to constructa probability current density which turned out to be neither conserved nor covariant. We resolvedthe apparent inconsistency of this observation with the physical requirement of the conservation andframe-independence of the total probability using a conserved and covariant current density. Theglobal conservation of the total probability is supported by the local conservation of this current den-sity. The latter is linked to a previously unnoticed global gauge symmetry of the Proca field with anAbelian gauge group. In Ref. [29], the authors have also attempted to give a spin-one generalizationof our treatment of the Klein-Gordon fields. However, because of various self-imposed restrictionsthey obtain a one-parameter subfamily of the inner products (( · , · )) { a } . This is related to the factthat the authors of [29] use the special Foldy transformation (65) (see Eq. (35) of [29]) to find themetric operator acting in the Hilbert space H ′ . Furthermore, in trying to impose the condition ofthe Lorentz-invariance of the metric operator, they transform the operator ˜ η + to a metric operatoracting in the Foldy representation and demand that the latter commutes with the generators of thePoincar´e group in this representation. This does not seem to be well-justified, because as seen from(40), the Hilbert space of the Foldy representation is just the direct sum of two copies of ˜ H . Hence,the metric operator associated with this representation is just the identity operator, not the onegiven by Eq. (39) of [29]. As seen from (117), this is the Foldy transformation which depends onthe choice of the unknown parameters appearing in the operator ˜ η + , not the metric operator of theFoldy representation. We would also like to stress that the analysis of [29] does not include theconstruction of the observables of the system or any treatment of its physical aspects such as thenotorious problem of the probabilistic interpretation of the quantum mechanics of Proca fields.Apart from the historical importance of the subject, the present work is mainly motivated bythe close analogy of the Proca and Maxwell fields. Performing the zero-mass limit of our resultsin an appropriate manner should lead to a consistent quantum mechanical treatment of individualphotons. As it is to be expected, there are subtleties in performing this limit. Nevertheless, we havebeen able to make some progress toward solving the problem of the construction of the Hilbert spaceand observables for the photon. We plan to report the results in a separate article. Acknowledgment
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