Quantum Mechanics of Proca Fields

TLDR: In this article, the most general physically admissible positive-definite inner product on the space of Proca fields was constructed, and a five-parameter family of Lorentz invariant inner products were used to construct a genuine Hilbert space for the quantum mechanics of the Proca field.

Abstract: We construct the most general physically admissible positive-definite inner product on the space of Proca fields. Up to a trivial scaling this defines a five-parameter family of Lorentz invariant inner products that we use to construct a genuine Hilbert space for the quantum mechanics of Proca fields. If we identify the generator of time-translations with the Hamiltonian, we obtain a unitary quantum system that describes first-quantized Proca fields and does not involve the conventional restriction to the positive-frequency fields. We provide a rather comprehensive analysis of this system. In particular, we examine the conserved current density responsible for the conservation of the probabilities, explore the global gauge symmetry underlying the conservation of the probabilities, obtain a probability current density, construct position, momentum, helicity, spin, and angular momentum operators, and determine the localized Proca fields. We also compute the generalized parity ($\cP$), generalized time-reversal ($\cT$), and generalized charge or chirality ($\cC$) operators for this system and offer a physical interpretation for its $\cP\cT$-, $\cC$-, and $\cC\cP\cT$-symmetries.

Figures

FIG. 1. Plots of I1, I2, and I3 in terms of the radial distance x−y . The radial distance is scaled with the Compton wave length M−1.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Quantum mechanics of proca fields" ?

In this paper, the authors have used the methods of PHQM to devise a complete formulation of the RQM of the Proca fields that does not involve restricting to the positive energy solutions of the proca equation. 

Q2. What are the future works in "Quantum mechanics of proca fields" ?

The authors plan to report the results in a separate article. 

Q3. How can the authors construct a faithful representation of the group G a?

The authors can construct a faithful representation of the group G a using the six-component representation A= A+,+1 ,A+,−1 ,A+,0 ,A−,+1 ,A−,−1 ,A−,0 T, where C and H are, respectively, represented by 3 and 12. 

Q4. What are the two Hilbert operators used to obtain the explicit form of the localized Proca?

The authors can use 148 and 149 , and the unitary operator 117 to obtain the explicit form of the localized Proca fields and the physical observables acting in H a . 

Q5. What is the corresponding state vector of a, s3 and s12?

The corresponding state vectors z , ,s are defined as the common eigenvectors of a, s3, and s12, i.e., a z , ,s =z z , ,s and s3 z , ,s = z , ,s , s12 z , ,s =s z , ,s , where z C3, −, + , and s −1,0 , +1 . 

Q6. how do the authors calculate the probability density of a position measurement?

For a position measurement to be made at time t=x0 /c, the authors have the probability densityx0,x = 2M UA x0,x 2 + UD−1/2Ȧ x0,x 2 = 2M UA x 2 + UAc x 2 . 197The authors can use the method discussed in Sec. V to introduce a current density J such that J0 = . 

Q7. What is the axiomatic approach to the position operator x0?

186In addition to being a Hermitian operator acting in the physical Hilbert space H, the position operator x0 has the following notable properties. 

Q8. What is the general invariant positivedefinite inner product?

The most general invariant positivedefinite inner product corresponds, therefore, to the most general biorthonormal system that consists of the eigenvectors of H and H†. 

Q9. What is the simplest way to determine the action of the position and spin operators on a?

the authors evaluate the action of the momentum, angular momentum, and helicity operators on A. Because P0 and commute, in view of 148 and 59 , the authors have p0=Ux00 −1P0 Ux0 0. 

Q10. What is the simplest way to show that the G a group is a global gauge?

It is not difficult to show that G a is a compact subgroup of this group and consequently isomorphic to U 1 if and only if all the parameters a ,h are rational numbers, otherwise G a is isomorphic to R+.Clearly, the G a gauge symmetry associated with the conservation of the total probability is a global gauge symmetry. 

Q11. What is the simplest way to define a unitary operator?

As seen from 54 , the operator Ux0 for any value of x0 R is a unitary operator mapping H to K. Following Refs. 26 and 27 the authors can use this unitary operator to define a Hamiltonian operator h acting in H that is unitary equivalent to H. Let x00 R be an arbitrary initial x0, and h:H→H be defined byh ª 

Q12. How do the authors find the probability of the localization of a Proca field in a?

As in nonrelativistic QM, the authors identify the probability of the localization of a Proca field A in a region V R3, at time t0=x0 0 /c, withPV = Vd3x ! 

Q13. What is the correct nonrelativistic limit for the spin operator s0?

the authors can obtain the explicit form of the spin operator s0 acting on A H. Again, noting that S ªs0A is a three-component field whose components satisfy 11 and 13 , the authors can determine S in terms of the initial data S x0 0 ,S˙ x0 0 .