Maxwell quantum mechanics

TL;DR

This paper develops a first quantized, Lorentz-covariant quantum theory of the photon based on Maxwell's classical fields, defining a positive-definite photon number density and a Hermitian position operator.

Contribution

It introduces a novel photon wave function with a conserved four-current, localized eigenvectors, and maintains covariance and causality within a quantum framework.

Findings

Photon position operator is Hermitian with localized eigenvectors.

Photon probability amplitude is the real part of the projection onto position eigenvectors.

The theory ensures causal propagation and zero net energy absorption without charged matter.

Abstract

We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive energy photon. With invariant plane wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.

Equations

$$\square \equiv {\partial }_{\mu }{\partial }^{\mu }={\partial }_{ct}^2-{\nabla }^2,$$

$$\square A^{\nu }=0.$$

$${\pi }^{\nu }=-{ϵ}_0{\partial }_t{A}^{\nu }.$$

$${\hat{A}}^{\mu }\left(x\right)$$

$${\hat{A}}_{\lambda }^{\mu }\left(x\right)$$

$$+{\hat{a}}_{\lambda }^{†}\left(\mathbf{k}\right)e_{\lambda }^{\mu \ast }{e}^{\mathrm{i}kx}]$$

$${\hat{\pi }}_{\lambda }^{\mu }\left(x\right)=-{ϵ}_0{\partial }_t{\hat{A}}_{\lambda }^{\mu }\left(x\right)$$

$$e_0=n^{\mu }=\left(1,0,0,0\right),{\mathbf{e}}_3\left(\mathbf{k}\right)={\mathbf{e}}_{\mathbf{k}}=\mathbf{k}/\left|\mathbf{k}\right|,$$

$${\mathbf{e}}_{\lambda }^{\ast }\left(\mathbf{k}\right)\cdot {\mathbf{e}}_{{\lambda }^{\prime }}\left(\mathbf{k}\right)={\delta }_{\lambda ,{\lambda }^{\prime }}\text{ for}\lambda ,{\lambda }^{\prime }=1,2,3.$$

$$\zeta =\left(-1,1,1,1\right)$$

$$e_{\lambda \mu }^{\ast }{e}_{{\lambda }^{\prime }}^{\mu }=e_{\lambda 0}^{\ast }{e}_{{\lambda }^{\prime }}^0-{\mathbf{e}}_{\lambda }^{\ast }\cdot {\mathbf{e}}_{{\lambda }^{\prime }}=-{\zeta }_{\lambda }{\delta }_{\lambda ,{\lambda }^{\prime }}.$$

$$\left[{\hat{a}}^0\left(\mathbf{k}\right)-{\hat{a}}^3\left(\mathbf{k}\right)\right]|0⟩=0.$$

$$\left[{\hat{a}}_{\lambda }\left(\mathbf{k}\right),{\hat{a}}_{{\lambda }^{\prime }}\left(\mathbf{q}\right)\right]$$

$$\left[{\hat{a}}_{\lambda }\left(\mathbf{k}\right),{\hat{a}}_{{\lambda }^{\prime }}^{†}\left(\mathbf{q}\right)\right]$$

$$|1_{\mathbf{k}\lambda }⟩={\hat{a}}_{\lambda }^{†}\left(\mathbf{k}\right)|0⟩$$

$$⟨1_{\mathbf{k}\lambda }|1_{\mathbf{q}{\lambda }^{\prime }}⟩={\delta }_{\lambda ,{\lambda }^{\prime }}{\left(2\pi \right)}^{3}2{\omega }_k\delta \left(\mathbf{k}-\mathbf{q}\right).$$

$$\left[{\hat{A}}_{\lambda }^{\mu }\left(t,\mathbf{x}\right),{\hat{A}}_{{\lambda }^{\prime }\mu }\left(t,\mathbf{y}\right)\right]$$

$$\left[{\hat{A}}_{\lambda }^{\mu }\left(t,\mathbf{x}\right),{\hat{\pi }}_{{\lambda }^{\prime \prime }\mu }\left(t,\mathbf{y}\right)\right]$$

$$|A_{x\lambda }^{\mu }⟩=\sqrt{\frac{\hslash }{{ϵ}_0}}\int \frac{\mathrm{d}\mathbf{k}}{{\left(2\pi \right)}^{3}2{\omega }_k}{e}^{-\mathrm{i}kx}{e}_{\lambda }^{\mu }\left(\mathbf{k}\right)|1_{\mathbf{k}\lambda }⟩.$$

$$⟨A_{t,\mathbf{x}}^{\mu }|{\pi }_{t,\mathbf{y}\mu }⟩-⟨{\pi }_{t,\mathbf{y}\mu }|A_{t,\mathbf{y}}^{\mu }⟩=-\mathrm{i}\hslash {\zeta }_{\lambda }{\delta }_{\lambda ,{\lambda }^{\prime }}\delta \left(\mathbf{x}-\mathbf{y}\right).$$

$$|{\pi }_{x\lambda \mu }⟩\equiv {\hat{\pi }}_{\lambda \mu }\left(x\right)|0⟩.$$

$$⟨A_{t,\mathbf{x}}^{\mu }|{\pi }_{t,\mathbf{y}\mu }⟩$$

$$=-\mathrm{i}\hslash \frac{1}{8{\pi }^{2}r}\frac{\partial }{\partial r}\sum _{\gamma =\pm }$$

$$\times \left[\pi \delta \left(r-\gamma ϵct\right)+\mathrm{i}\gamma PV\left(\frac{1}{r-\gamma ϵct}\right)\right].$$

$$⟨A_{t,\mathbf{x}}^{\mu }|{\partial }_{ct}{A}_{t,\mathbf{y}\mu }⟩-⟨{\partial }_{ct}{A}_{t,\mathbf{y}\mu }|A_{t,\mathbf{x}}^{\mu }⟩=\mathrm{i}\frac{\hslash }{2{ϵ}_{0}c}{\zeta }_{\lambda }{\delta }_{\lambda ,{\lambda }^{\prime }}\delta \left(\mathbf{x}-\mathbf{y}\right).$$

$$\left|\mathbf{k}\right|A_{\parallel }=\left({\omega }_k/c\right)A^0$$

$$A_{\lambda }^{ϵ\mu }\left(t,\mathbf{x}\right)=\mathrm{i}\sqrt{\frac{\hslash }{{ϵ}_0}}\int \frac{\mathrm{d}\mathbf{k}}{{\left(2\pi \right)}^{3}2{\omega }_k}{c}_{\lambda }^{ϵ}\left(\mathbf{k}\right)e_{\lambda }^{\mu }\left(\mathbf{k}\right)e^{-iϵ\left({\omega }_k-\mathbf{k}\cdot \mathbf{x}\right)}.$$

$$A_c\equiv \mathrm{i}{\hat{D}}^{-1/2}{\partial }_{ct}A$$

$$\hat{D}\equiv -{\nabla }^2.$$

$$\mathrm{i}{\hat{D}}^{-1/2}{\partial }_{ct}{A}^{ϵ}=ϵA^{ϵ}.$$

$$\mathrm{i}g{\partial }_{\mu }{A}_{\nu }^{\ast }\left(x\right){\overleftrightarrow{\partial }}^{\mu }{\tilde{A}}^{\nu }\left(x\right)=0$$

$$f{\overleftrightarrow{\partial }}_{\mu }g\equiv f\left({\partial }_{\mu }g\right)-g\left({\partial }_{\mu }f\right).$$

$${\partial }_t\rho +{\partial }_{\mathbf{x}}\cdot \mathbf{j}={\partial }_{\mu }{J}^{\mu }=0$$

Full text

I Introduction

One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function Sebens . The question addressed here and in Sebens is whether Maxwell’s equations can be interpreted as a first quantized theory of the photon. We answer this question in the affirmative and show that classical electromagnetism in the Lorenz gauge can be extended to a quantum mechanical theory that satisfies all of the usual rules. A first quantized theory of a relativistic particle should have a Lorentz covariant wave equation, a scalar lagrangian, future orientation and causal propagation Kazemi ; Kiessling . In addition it should have a conserved probability current that transforms like a Lorentz vector and Hermitian operators representing observables that are not part of classical field theory.

We waive the conditions of complex fields and a first order wave equation stated in Kiessling since, according to the Hegerfeldt theorem Hegerfeldt , this is incompatible with causal propagation. Hegerfeldt proved that any positive frequency field localized for an instant spreads instantaneously to give a field that is nonzero throughout space. In classical electromagnetic theory fields are real and propagate causally, and in relativistic quantum mechanics the fields describing neutral bosons are real ItzyksonZuber ; MandlShaw . Our solution to the causality problem is to formulate photon quantum mechanics in terms of real fields. This requires a number density that is positive definite for both positive and negative frequency fields. Such an inner product does indeed exist, and its properties were investigated by Mostafazadeh and co-workers MostafazadehZamani ; BabaeiMostafazadeh . Here we extend their photon density to a four-current that transforms as a Lorentz vector. We interpret positive frequency as absorption of positive energy particles and negative frequency as emission of positive energy particles; negative energy photons or retrocausal propagation are not required.

An important contribution to the development of photon quantum mechanics is investigation of the simpler Klein-Gordon equation describing neutral pions. With the exception of the work of Mostafazadeh and co-workers, this has been largely based on positive frequency solutions to a wave equation that is a first order in time Kazemi ; RosenHorwitz ; Sutherland with position eigenvectors that are nonlocal and noncovariant NewtonWigner ; RosenHorwitz or nonorthogonal Phillips ; Sutherland . An exception is Wharton’s formulation that imposes initial and final conditions Wharton . Work on the photon has followed a different but sometimes interwoven path that was recently reviewed by Sebens Sebens and by Kiessling and Tarvildar-Zadeh Kiessling in their Appendix A. They propose a first order Dirac-like equation for a single free photon and show that it satisfies all of the requirements outlined at the end of paragraph one.

Up to the turn of this century there was no photon number density and no true photon quantum mechanics. This is a consequence of several obstacles to its development: Newton and Wigner NewtonWigner derived position operators for electrons and for spin 0 Klein-Gordon particles but found that, if invariance under rotations is assumed, no photon position operator exists. Even these Klein-Gordon position eigenvectors are problematic since they are not Lorentz covariant and they are nonlocal in configuration space. Pryce Pryce derived a photon position operator but its components do not commute, so it does not define a basis of position eigenvectors. The standard density is positive only if the fields are restricted to positive frequencies however, as discussed in a previous paragraph, positive frequency fields alone do not propagate causally Hegerfeldt . Some form of a photon wave function is thought to be useful in quantum optics but, as a consequence of the above mentioned obstacles, the usual rules of quantum mechanics were not followed. Instead the Weber vector Weber ; Kiessling discussed by Bialynicki-Birula BB96 and the Sipe photon wave function Sipe are based on energy density SmithRaymer . Alternatively the Weber vector can be divided by the square root of frequency to give number amplitude Good . This wave function was examined recently by Sebens Sebens , but found to be unsatisfactory because probabilities do not always transform properly under Lorentz transformations.

The question of localizability is fundamental to quantum field theory (QFT) where particles interact locally at a common space-time coordinate. Particle position, while essential to this picture, is especially problematic in the case of photons that have no rest frame. Recently, derivation of a photon position operator with commuting components HawtonPosOp ; HawtonBaylis and the Mostafazadeh particle density that is positive definite for both positive and negative frequency waves MostafazadehZamani have led to photon quantum mechanics HawtonDebierre ; BabaeiMostafazadeh with a Hilbert space consisting of solutions to Maxwell’s equations and an invariant inner product. However, this is not an entirely satisfactory first quantized theory since in BabaeiMostafazadeh the position operator and its eigenvectors are of the noncovariant Newton-Wigner form, while in HawtonDebierre the position operator is not Hermitian and biorthogonality is derived from QFT. Also, in BabaeiMostafazadeh ; HawtonDebierre and in Sebens , no photon density that transforms like a Lorentz vector was found. Here we derive a photon number density that transforms as the zeroth component of a Lorentz vector. In addition, we show that invariant plane wave normalization leads to a Hermitian photon position operator with localized eigenvectors that transform as Lorentz vectors.

In the next Section we give a brief summary of relativistic notation and some material from the QFT text book ItzyksonZuber that motivated our work. Since we discuss only photons and Fermions, quantum electrodynamics (QED) as discussed in QFT text books is sufficient for our purposes. In Section III we write the general solution to Maxwell’s wave equation as a sum over positive and negative frequency terms ($\epsilon=\pm$) in the Lorenz gauge, define the Babaei-Mostafazadeh conjugate field, and derive a photon four current. An invariant inner product is then defined in cofiguration space and in momentum space. In Section III.B the photon position operator with commuting components is reviewed. It is found that its eigenvectors are localized Lorentz vectors. An expression for probability amplitude is obtained by projecting onto a basis of position eigenvectors and taking its real part. SI units are used throughout.

II Background: Relativity and QFT

II.1 Relativistic notation

In relativity the space-time coordinates are incorporated into the 4-vector $x=x^{\mu}=\left(ct,\mathbf{x}\right)$ with Lorentz invariant length squared $x^{2}=x_{\mu}x^{\mu}$ where $x_{\mu}=g_{\mu\nu}x^{\nu}$ and $g_{\mu\nu}=g^{\mu\nu}$ is a $4\times 4$ matrix with diagonal $\left(1,-1,-1,-1\right)$. The classical (in the non-quantum sense) energy-momemtum four-vector is $p^{\mu}=\left(E_{p}/c,\mathbf{p}\right)=\hbar\left(\omega_{k}/c,\mathbf{k}\right)$ where we define $\omega_{k}$ to be positive. The four-potential is $A^{\mu}=\left(\phi/c,\mathbf{A}\right)$ where $\phi$ and $\mathbf{A}$ are the scalar and vector potential respectively. In first quantization energy and momentum are replaced with operators, that is $E_{p}\rightarrow$i$\hbar\partial_{t}$ and $\mathbf{p}\rightarrow\widehat{\mathbf{p}}=-$i$\hbar\partial_{\mathbf{x}}$. Here $E_{p}$ is the energy of a particle with momentum $\mathbf{p},$ $c$ is the speed of light, $h=2\pi\hbar$ is Planck’s constant, $\partial_{t}\equiv\partial/\partial t$ and $\partial_{\mathbf{x}}\equiv\nabla$. Some additional invariants that will be used here are $p_{\mu}p^{\mu}=E_{p}^{2}/c^{2}-\mathbf{p}^{2}c^{2}$,

[TABLE]

$kx=\omega_{k}t-\mathbf{k\cdot x}$ and $\exp\left(-\mathrm{i}kx\right)$. The $\mathbf{k}$-space Lorenz gauge condition, $kA=0$, is Lorentz invariant while the Coulomb gauge condition, $\mathbf{k}\cdot\mathbf{A}=0$, is not so, in keeping with our theme of maintaining manifest covariance, we will use the Lorenz gauge. Operation with $\widehat{E}_{p}^{2}-\widehat{\mathbf{p}}^{2}c^{2}$ on the four-potential $A\left(x\right)=A\left(t,\mathbf{x}\right)$ gives the Maxwell wave equation,

[TABLE]

Since this equation is second order in both space and time and $m=0$, it has the remarkable property that it is unchanged by multiplication by $\hbar^{2}$ so it can be interpreted as a classical or a quantum wave equation.

II.2 Lessons from QFT text books

Because QFT is based on local interactions it is common in QFT to use the Lorenz gauge and the Gupta-Bleuler indefinite metric GuptaBleuler ; MandlShaw ; ItzyksonZuber ; HawtonWaveFunctions99 . This approach is covariant and based on four essentially identical copies of the solution to the $m=0$ Klein-Gordon wave equation $\square\phi=0$. The Lagrangian is the usual one plus a term proportional to the square of the Lorenz gauge condition, $\mathcal{L=-}\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\mu_{0}}\left(\partial_{\mu}A^{\mu}\right)^{2}$ where $\mu_{0}$ is the magnetic permeability of vacuum and $F^{\mu\nu}$ is the electromagnetic tensor whose first row is $F^{0\nu}=\left(0,\mathbf{E}\right)/c$, $\mathbf{E}$ being the electric field vector. The field conjugate to $A_{\nu}$ is $\partial\mathcal{L}/\partial\left(\partial_{ct}A_{\nu}\right)=F^{0\nu}-g^{0\nu}\left(\partial_{\mu}A^{\mu}\right)=\pi^{\nu}$ with

[TABLE]

In a source free region the Lagrange equation of motion is just Maxwell’s wave equation, (2).

The second quantized four-potential operator can be written as ItzyksonZuber

[TABLE]

where $\widehat{a}_{\lambda}\left(\mathbf{k}\right)$ and $\widehat{a}_{\lambda}^{\dagger}\left(\mathbf{k}\right)$ are annihilation and creation operators for a photon with wave vector $\mathbf{k}$ and polarization $\lambda$ and $e_{\lambda}^{\mu}\left(\mathbf{k}\right)$ is a unit vector. The first term in (4) annihilates a photon at $x$, while the second term creates one. The conjugate field operator is

[TABLE]

With the polarization unit vectors defined such that 0 is time-like, 1 and 2 are transverse and 3 is longitudual,

[TABLE]

[TABLE]

Defining

[TABLE]

this can be written as

[TABLE]

The operator version of the Lorenz gauge condition is then

[TABLE]

In the absence of an electric four-current density the number of longitudinal photons must equal the number of scalar photons. The creation and annihilation operators for a photon with wave vector $\mathbf{k}$ and helicity $\lambda$ will be assumed to satisfy the invariant commutation relations

[TABLE]

so that the one photon states

[TABLE]

are invariantly normalized according to

[TABLE]

An annihilation operator acting on the vacuum state gives zero, that is $\widehat{a}_{\lambda}\left(\mathbf{k}\right)\left|0\right\rangle=0$ and $\widehat{A}_{\lambda}^{\mu+}\left(x\right)\left|0\right\rangle=0$. It can be proved using (4), (5), (9), and (11) that the $\mathbf{x}$-space photon commutation relations at time $t$ are

[TABLE]

where summation over $\mu$ is implied. The factor $\zeta_{\lambda}$ comes from (9). It was omitted in the definition (11) to avoid introducing it twice. A creation operator at $x$ acting on the vacuum state gives the 1-particle state

[TABLE]

The vacuum expectation value of (14) can be written as

[TABLE]

with

[TABLE]

Creation at $\left(t,\mathbf{y}\right)$ followed by annihilation at $\left(t,\mathbf{x}\right)$ and creation at $\left(t,\mathbf{x}\right)$ followed by annihilation at $\left(t,\mathbf{y}\right)$ are equally likely. This implies that we don’t know whether the photon was created or destroyed at $\left(t,\mathbf{x}\right)$.

We will illustrate Hegerfeldt instantaneous spreading by evaluating the positive frequency part of (16). Integration of $\left\langle A_{t,\mathbf{x}}^{\mu}|\pi_{\mu t,\mathbf{y}}\right\rangle$ after substitution of (15), (17) and (13) with $\mathbf{r\equiv x}-\mathbf{y}$ and $t\equiv t_{x}-t_{y}$ gives HawtonDebierre

[TABLE]

At $t=0$ the principal value ($PV$) terms cancel while the $\delta$-functions add to give a factor $2$. The $1/r$ tails of the $PV$ terms are instantaneously masked by destructive interference Karpov , only their sum is localized. Eq. (18) was included to illustrate the nonlocality of the $PV$ terms but, since $\left\langle\pi_{t,\mathbf{y}\mu}|A_{t,\mathbf{y}}^{\mu}\right\rangle=-\left\langle A_{t,\mathbf{x}}^{\mu}|\pi_{t,\mathbf{y}\mu}\right\rangle^{\ast}$, (16) that is a sum of positive and negative frequency terms can be written as

[TABLE]

To sum up, the photon equation of motion in the Lorenz gauge can be derived from a scalar Lagrangian. Time-like, longitudinal, and transverse unit vectors were defined. Derivation of the configuration space commutation relations (14) illustrates the role of invariant normalization of the plane wave basis. The QFT commutation relations that ensure microcausality require a sum of a term describing propagation from $y$ to $x$ and a term representing propagation from $x$ to $y$, giving a real result. This implies that emission at $y$ followed by absorption at $x$ and emission at $x$ followed by absorption at $y$ are equally likely; we don’t know if a the photon was emitted or absorbed at $x$.

III Maxwell quantum mechanics

III.1 Hilbert space

A first quantized theory requires a vector space, a conserved four-current, an inner product, a positive definite norm, and operators representing observables. Here the photon vector space consists of solutions to the Maxwell wave equation (2) subject to the Lorenz gauge condition $\partial_{\mu}A^{\mu}=0$. In $\mathbf{k}$-space the Lorenz gauge condition is

[TABLE]

so, since $\omega_{k}=c\left|\mathbf{k}\right|$ in vacuum, the probability amplitude for a longitudinal mode equals the probability amplitude for a scalar mode. Since (2) is second order in time, completeness requires both positive and negative frequency solutions, that is $A^{\mu}\left(t,\mathbf{x}\right)=\sum_{\lambda,\epsilon=\pm}A_{\lambda}^{\epsilon\mu}\left(t,\mathbf{x}\right)$ where

[TABLE]

The unit imaginary i was introduced for later convenience to give the integrand $\sin\left(kx+\arg c^{+}\right)$ that is an odd function of $kx$ if $c_{\lambda}^{\epsilon}\left(\mathbf{k}\right)$ is real. The $\epsilon=+$ terms $e^{-\mathrm{i}\omega_{k}t}$ are referred to as positive frequency while the $\epsilon=-$ terms $e^{\mathrm{i}\omega_{k}t}$ are called negative frequency. Our sign convention is that positive $\epsilon$ signifies absorption and negative $\epsilon$ denotes emission which can be thought of as negative or time reversed absorption. Eq. (15) is the state vector describing a positive frequency one photon state in this vector space, but there are no particle annihilation and creation operators in a first quantized theory, so it cannot be attributed to the action of a creation operator on the vacuum.

Babaei and Mostafazadeh BabaeiMostafazadeh define the conjugate field****

[TABLE]

where

[TABLE]

It can be verified by substitution that if $A$ satisfies Maxwell’s wave equation, $A_{c}$ also satisfies this equation. The operator on the right hand side of (22) extracts the sign of the photon frequency, $\epsilon$, according to

[TABLE]

The inner product on a $t$-hyperplane should equal the spatial integral of the $0^{th}$ component of a four-current, so the photon continuity equation will be considered next. For any $A$ and $\widetilde{A}$ that satisfy (2), a continuity equation can be obtained by subtracting $\widetilde{A}$ multiplied by the Maxwell wave equation for $A^{\ast}$ from $A^{\ast}$ times the Maxwell wave equation for $\widetilde{A}.$ This is the method used to derive the Schrödinger continuity equation in undergraduate quantum mechanics. After cancellation of the $\partial_{\mu}A_{\nu}^{\ast}\partial^{\mu}\widetilde{A}^{\nu}$ terms, substitution of (2) and multiplication by the constant i$g$,

[TABLE]

where

[TABLE]

Eq. (25) is of the form

[TABLE]

where $\rho$ is density, $\mathbf{j}$ is current density and $J^{\mu}$ is four-current density. If $\widetilde{A}=\phi$ and $g=$ $e\epsilon_{0}c/\hbar$ where $e$ is the magnitude of the charge on the electron, $\rho$ is pion charge density. While this charge density is not directly relevant to the photon in vacuum, it provides a simple argument for selecting $g=\epsilon_{0}c/\hbar$ to describe number density. If $\widetilde{A}=A_{c}$ (25) is a continuity equation for four-current density. The choice $g=\epsilon_{0}c/\hbar$ in (25) is also consistent with the QFT expression (19) that describes the orthogonality of the localized photon states at $\left(t,\mathbf{x}\right)$ and $\left(t,\mathbf{y}\right)$.

With $g=\epsilon_{0}c/\hbar$ and $\widetilde{A}=A_{c}$ the photon four-current (25) becomes

[TABLE]

where $A$ and $A_{c}$ satisfy (2). The minus sign in (28) ensures that its space-like terms are positive while its time-like term is negative, so that the scalar and longitudinal terms make no net contribution to $J^{0}\left(x\right)$.

In the basis of positive and negative frequency definite helicity states, (21), substitution of (22) in (28) gives

[TABLE]

where we have used the fact that the two terms in (28) are equal if $\epsilon^{\prime}=\epsilon$ and cancel if $\epsilon^{\prime}=-\epsilon$. From (24) $\mathrm{i}\partial_{ct}A^{\epsilon}\equiv\epsilon\widehat{D}^{-1/2}A^{\epsilon}$, so (29) is a sum of positive and negative frequenc$\epsilon$y terms, in contrast to the standard (charge) density that is their difference.

The photon inner product on a hyperplane of simultaneity at instant $t$ will be defined as

[TABLE]

It can be generalized to the covariant form

[TABLE]

on an arbitrary spacelike hyperplane with normal $n_{\mu}$ MostafazadehZamani ; HawtonDebierre . The choice $n=\left(1,0,0,0\right)$ gives (30).

Substitution of (22) and the invariant plane wave normalization (13) in (30) with $c\widehat{D}^{1/2}\rightarrow\omega_{k}$ gives

[TABLE]

The integrand of (32) is just photon density in the form (29) if $A_{2\lambda^{\prime}}^{\epsilon^{\prime}}=A_{1\lambda}^{\epsilon}=A_{\lambda}^{\epsilon}$. In the second line the sum over $\nu$ has been evaluated to give the factor $-\delta_{\lambda,\lambda^{\prime}}\zeta_{\lambda}$ that follows from (9).

The configuration space four-potential (21) can be generalized to a state vector in an arbitrary basis in a manner analogous to the vector notation used in three dimensions CT . Eq. (21) is isomorphic to the state vector

[TABLE]

Substitution of (34) in (33) with $c\widehat{D}^{1/2}\rightarrow\omega_{k}$ gives

[TABLE]

With the more common non-invariant plane wave normalization denoted by the subscript NW (Newton-Wigner),

[TABLE]

the $\mathbf{k}$-space inner product that replaces (35) is

[TABLE]

All modes of the four-potential can be treated similarly except that $A^{0}$ makes a negative contribution to the inner product, while its spatial components make positive contributions. Since (10) and (20) imply equal numbers of scalar and longitudinal photons, their contributions cancel. States $\left|n\right\rangle$ containing no transverse modes but only $n$ scalar/longitudual pairs are equivalent to the vacuum so their contribution to the Fock space inner product is $\left\langle n|n\right\rangle=\left\langle 0|0\right\rangle$ ItzyksonZuber . The one-photon inner product $\left(A_{1\lambda}^{\epsilon},A_{2\lambda^{\prime}}^{\epsilon^{\prime}}\right)$ does not count these scalar/longitudual pairs and contains no vacuum contribution so only transverse photons are counted.

For transverse modes the boundary condition at $t_{0}$ can be written as

[TABLE]

where $\mathbf{E}_{\lambda}\left(t,\mathbf{x}\right)=-\partial_{t}\mathbf{A}_{\lambda}\left(t,\mathbf{x}\right)$ for $\lambda=1,2$. This is analogous to imposing initial conditions on the displacement and velocity of an infinitesiment element of a vibrating string or membrane.

III.2 Photon observables

In this subsection we will find the eigenvectors and eigenvalues of the momentum, energy, position, and angular momentum operators and write the position space probability amplitude as the projection of an arbitrary state vector onto bases of position eigenvectors and obtain real functions by taking the real parts of the $\epsilon=+$ fields and probability amplitude. Examples of circular and linear polarization, plane waves, and propagation toward and away from a point of localization are given.

In $\mathbf{k}$-space the momentum operator is $\widehat{\mathbf{P}}=\hbar\mathbf{k}$ and the photon position operator with commuting components, $\widehat{\mathbf{x}}$, is related to the position operator $\mathrm{i}\mathbf{\partial}_{\mathbf{k}}$ as $\widehat{\mathbf{x}}=\widehat{R}k^{\alpha}\mathrm{i}\mathbf{\partial}_{\mathbf{k}}k^{-\alpha}\widehat{R}^{-1}$ where HawtonBaylis

[TABLE]

The operator $\widehat{R}$ rotates $\mathbf{e}_{1}+\mathrm{i}\sigma\mathbf{e}_{2}$ about $\mathbf{e}_{2}$ by $\theta$, then about $\mathbf{e}_{3}$ by $\phi$ to give $\mathbf{e}_{\theta}+\mathrm{i}\sigma\mathbf{e}_{\phi}$, and finally about $\mathbf{e}_{\mathbf{k}}$ by $\chi$. Here $\partial_{\mathbf{k}}$ is the $\mathbf{k}$-space gradient, $\widehat{S}_{i}$ are the Cartesian components of the spin operator $\widehat{\mathbf{S}}$,

[TABLE]

is the helicity operator, $\theta$ and $\phi$ are the $\mathbf{k}$-space spherical polar angles, $\chi\left(\theta,\phi\right)$ is the Euler angle and the $\mathbf{k}$-space spherical polar unit vectors are $\mathbf{e}_{\theta},$ $\mathbf{e}_{\phi}$ and $\mathbf{e}_{\mathbf{k}}$. The transverse unit vectors $\widehat{R}\left(\mathbf{e}_{1}+\mathrm{i}\lambda\mathbf{e}_{2}\right)/\sqrt{2}$ with helicity $\lambda=\pm 1$ are

[TABLE]

The longitudinal unit vector is $R\mathbf{e}_{3}=\mathbf{e}_{\mathbf{k}}$ and the scalar unit vector that is unrotated by $R$ is given by (6).

The photon position operator with commuting components derived in HawtonPosOp is

[TABLE]

The parameter $\alpha$ was introduced for convenience so that both Newton-Wigner $\left(\alpha=1/2\right)$ and covariant $\left(\alpha=0\right)$ position eigenvectors could be included in a single expression. If $\alpha=1/2$, (43) equals the Pryce position operator plus a term proportional to

[TABLE]

It is this additional term that gives $\widehat{\mathbf{x}}^{\left(\alpha\right)}$ commuting components. The Euler angle $\chi\left(\theta,\phi\right)$ is defined as a general rotation about $\mathbf{k}$. Any possible transverse basis is the set of eigenvectors of (43) for some $\chi\left(\theta,\phi\right)$. Since experiments are often performed on optical beams with definite angular momentum, the case $\chi=-m\phi$ for which the position eigenvectors have intrinsic angular momentum $\hbar m\sigma$ in some arbitrary but fixed direction is of special interest. For this choice of $\chi$ (44) becomes

[TABLE]

An alternative to specifying the $t$-hyperplane is to work in the Schrödinger picture where operators and their eigenvectors are time independent and the time dependence is contained in the state vector SakuraiModernQM . Transformation between the Schrödinger and Heisenberg pictures can be performed using a unitary time evolution operator. In the $\epsilon=\pm$ basis time evolution is determined by i$\partial_{t}A_{\lambda}^{\epsilon}\left(x\right)=\epsilon c\widehat{D}^{1/2}A_{\lambda}^{\epsilon}\left(x\right)$ that can be integrated from $t$ to $t+\tau$ to give

[TABLE]

where $\widehat{U}^{\epsilon}$ is a unitary operator satisfying $\widehat{U}^{\dagger}=\widehat{U}^{-1}$ so that $\widehat{U}\widehat{U}^{\dagger}=\widehat{U}^{\dagger}\widehat{U}=\widehat{1}$.

The operator equation for a state vector with momentum $\hbar\mathbf{q}$ and helicity $\lambda$ is

[TABLE]

These states will be normalized according to (13). The basis of momentum eigenvectors is orthogonal, Dirac $\delta$-function normalized and complete CT . Thus any state $\left|A\right\rangle$ can be described by an integral over these plane waves. The Hamiltonian operator

[TABLE]

has positive energy eigenvectors, $\hbar c\left|\mathbf{k}\right|=\hbar\omega_{k}$.

The position space probability amplitude is the projection of a particle’s state vector onto a basis of position eigenvectors. In the Schrödinger picture the $\mathbf{k}$-space representation of the position operator is (43) and the eigenvector equation and position eigenvector at $\mathbf{y}$ are

[TABLE]

The choice $\alpha=0$ corresponds to the invariant normalization (13) with inner product (35), while the Newton-Wigner choice $\alpha=1/2$ requires (36) and (37). In either case, for $\lambda=\pm 1$,

[TABLE]

The factor $\delta\left(\mathbf{x}-\mathbf{y}\right)$ is infinite at $\mathbf{x}=\mathbf{y}$, that is the position eigenvectors, like the momentum eigenvectors, are not normalizable. The form of (35) is due to the invariant plane wave normalization (13) that introduces a factor $\omega_{k}$ into the inner product (33). Since the covariant case $\alpha=0$ is the focus of this paper, we will define $c_{\mathbf{y}\lambda}=c_{\mathbf{y}\lambda}^{\left(0\right)}$. In this case the Heisenberg picture $\mathbf{k}$-space position eigenvectors are the manifestly covariant Lorentz four-vectors

[TABLE]

Transformation to configuration space using the Lorentz invariant measure d$\mathbf{k}/k$ yields a four-vector proportional to the electromagnetic four-potential, (21). The inverse Fourier transform obtained using the trivial measure is the time derivative of the vector potential, proportional to the electric field describing this instantaneously localized position eigenvector. For definite helicity transverse modes $\lambda=\pm$, $\mathbf{B=-}\mathrm{i}\lambda\mathbf{E}$, so this is the Weber  vector, $\mathbf{E+}\mathrm{i}\lambda c\mathbf{B}=2\mathbf{E}$.

The eigenvalues are the possible observed values, so they must be real. Since $c_{\mathbf{x}\lambda}^{\left(\alpha\right)}$ and $c_{\mathbf{y\lambda}}^{\left(\alpha\right)}$ are eigenvectors of $\widehat{\mathbf{x}}^{\left(\alpha\right)}$ and $\widehat{D}^{-1/2}$ is Hermitian, the expectation value of the position operator gives

[TABLE]

If $\alpha=1/2,$ $\widehat{\mathbf{x}}^{\left(1/2\right)\dagger}=\widehat{\mathbf{x}}^{\left(1-1/2\right)}=\widehat{\mathbf{x}}^{\left(1/2\right)}$ is the well known Newton-Wigner result. If $\alpha=0$ the inner product is (35) and $\widehat{\mathbf{x}}^{\left(0\right)\dagger}=\widehat{\mathbf{x}}^{\left(0\right)}\equiv\widehat{\mathbf{x}}$. In either case $\widehat{\mathbf{x}}^{\left(\alpha\right)}$ is Hermitian. Since according to (52) $\left(A_{\mathbf{x\lambda}}^{\epsilon},A_{\mathbf{y\lambda}}^{\epsilon}\right)=\delta\left(\mathbf{x}-\mathbf{y}\right)$, it follows that

[TABLE]

that is the eigenvalues of $\widehat{\mathbf{x}}^{\left(\alpha\right)}$ are real as expected for Hermitian operators.

The state vector (34) can be projected onto the $\mathbf{k}$- space basis using (13). With $k=\left(\omega_{k}/c,\mathbf{k}\right)$ this gives

[TABLE]

The velocity operator

[TABLE]

equals $c\mathbf{e}_{k}$ in $\mathbf{k}$-space.

The transverse photon position eigenvectors have a definite three**-**component of total angular momentum with indefinite spin and orbital contributions. Writing the total angular momentum as the sum of its extrinsic and intrinsic parts,

[TABLE]

Using $\mathbf{e}_{\theta}\cdot\mathbf{e}_{3}=-\sin\theta$ and $\mathbf{e}_{\mathbf{k}}\cdot\mathbf{e}_{3}=\cos\theta$,

[TABLE]

In Cartesian components

[TABLE]

With $J_{3}=L_{3}+S_{3}$ where $L_{3}$ and $S_{3}$ are spin and intrinsic orbital angular momentum parallel to the $3$-axis, in the first term of (61) $L_{3}=\hbar\left(m\lambda+1\right)$ and $S_{3}=-\hbar\lambda,$ in the second $L_{3}=\hbar m\lambda$ and $S_{3}=0,$ while in the third term $L_{3}=\hbar\left(m\lambda-1\right)$ and $S_{3}=\hbar\lambda.$ In all terms their sum is $\hbar m\lambda$.

The position eigenvectors also form a basis where (52) is their closure relation CT . The four-potential and field obtained obtained by projection of (34) onto the basis of position eigenvectors $\left|1_{\lambda x}^{\epsilon\mu}\right\rangle$ is (21). Since for transverse modes $\mathbf{E}_{\lambda}\left(x\right)=-\partial_{t}\mathbf{A}_{\lambda}\left(x\right)$,

[TABLE]

for $\lambda=1,2$. The position space photon probability amplitude $\left(1_{\lambda x}^{\epsilon\mu},A\right)$ is

[TABLE]

According to the Hegerfeldt theorem positive frequency fields do not propagate causally. Eq. (18) in Section II.B is an example that illustrates this. As in classical electromagnetic theory it has been convenient up to this point to use $\epsilon=\pm$ basis, but we will now define the real fields and wave functions that propagate causally. For real fields $A_{\lambda}^{-\mu}\left(x\right)=A_{\lambda}^{+\mu\ast}\left(x\right)$ so $A_{\lambda}^{-\mu}\left(x\right)$ is redundant and we will define

[TABLE]

where $c_{\lambda}^{-}\left(\mathbf{k}\right)=0$ and $x=\left(t,\mathbf{x}\right)$.

Since $\mathbf{A}\left(x\right)$ is the indefinite time integral of $-\mathbf{E}\left(x\right)$, we will consider only a few examples of transverse $\mathbf{E}\left(x\right)$ and $\psi\left(x\right)$. Taking $c_{\lambda}^{+}\left(\mathbf{k}\right)=c_{\lambda_{0}}\left(\mathbf{k}\right)\delta_{\lambda,\lambda_{0}}$ where $c_{\lambda_{0}}\left(\mathbf{k}\right)$ is assumed to be real for simplicity,

[TABLE]

The field (68) is circularly polarized and rotates clockwise about $\mathbf{k}$ if $\lambda=+1$ and counterclockwise if $\lambda=-1$. If $c_{\lambda_{0}}\left(\mathbf{k}\right)=c_{\lambda_{0}}\delta\left(\mathbf{k}-\mathbf{q}\right)$ these are the circularly polarized plane waves,

[TABLE]

Circularly polarized states can be added and subtracted to give linearly polarization. If $c_{\lambda}^{\epsilon}\left(\mathbf{k}\right)=c_{\theta}\left(\mathbf{k}\right)\delta_{\epsilon,+}\left(\delta_{\lambda,+}+\delta_{\lambda,-}\right)/\sqrt{2}$ where $c_{\theta}\left(\mathbf{k}\right)$ is real,

[TABLE]

describes a linearly polarized photon. The choice $c_{\lambda}^{\epsilon}\left(\mathbf{k}\right)=c\left(\mathbf{k}\right)\delta_{\epsilon,+}\left(\delta_{\lambda,+}-\delta_{\lambda,-}\right)/\left(\mathrm{i}\sqrt{2}\right)$ gives

[TABLE]

If $c_{\theta}\left(\mathbf{k}\right)=c_{1}\delta\left(\mathbf{k}-q\mathbf{e}_{3}\right)$ and $c_{\phi}\left(\mathbf{k}\right)=c_{2}\delta\left(\mathbf{k}-q\mathbf{e}_{3}\right)$, these are the linearly polarized plane waves

[TABLE]

The plane wave states (69), (73) and (74) are not normalizable.

A photon localized at $\mathbf{y}$ on the $t_{0}=0$ hyperplane is described by $c_{\mathbf{y}\lambda}\left(\mathbf{k}\right)=e^{-\mathrm{i}\mathbf{k}\cdot\mathbf{y}}$ so that $\psi_{\mathbf{y}\lambda}\left(x\right)=\delta\left(\mathbf{x}-\mathbf{y}\right)$ at $t=0$. An explicit expression for its time evolution can be obtained by integrating (70) as in (18) and HawtonDebierre to give

[TABLE]

It is infinite at $r=\pm ct$ and not normalizable. Thus it does not allow calculation of a photon probability density, but it can be seen by inspection that it propagates causally, inward on a spherical shell if $t<0$ and outward if $t>0$.

Just as in Schrödinger quantum mechanics, the position and momentum eigenvectors are not square integrable but they form a basis and hence satisfy a completeness relation CT . Using (35), (67) and the Parserval-Plancherel identity the squared norm of the state vector $\left|A\right\rangle$ can be written as

[TABLE]

If $\left(A,A\right)$ is finite, $\left|A\right\rangle$ is normalizable and the $\mathbf{x}$-space probability density is

[TABLE]

This is the Born rule. In a normalized one photon state, $\left(A,A\right)=1$. The plane wave basis satisfies an analogous completeness relation that gives a $\mathbf{k}$-space density $\rho\left(k\right)$.

Babaei and Mostafazadeh defined a photon inner product BabaeiMostafazadeh that is closely related to (33). The parameter $g$, called $l$ in BabaeiMostafazadeh , is unspecified and they use a real symmetric/antisymmetric basis  written in terms of $\mathbf{A}$ and $\partial_{t}\mathbf{A}\propto\mathbf{E}$. Their position eigenvectors are of the Newton-Wigner form, the Coulomb gauge is used, and their photon number density is not the time-like component of a conserved four-current. They impose initial conditions on $\mathbf{A}$ and $\mathbf{E}$ as in (38) and (39) and derive a position operator that is the Heisenberg picture equivalent of (43).

IV Conclusion

In this paper we have derived a manifestly covariant quantum mechanical theory of the photon. The solutions to their equations of motion are real but, as in classical electromagnetic theory, we found it convenient to define the Hilbert space as a basis of positive and negative frequency waves using the Babaei-Mostafazadeh positive definite photon number density. These complex solutions can then be used to calculate real fields and wave functions as a final step. We extended the previous work by deriving a photon four-current in the Lorenz gauge with real potentials, fields and probability amplitude. We found for the first time that, with invariant plane wave normalization, the position operator is Hermitian and orthogonality of its covariant eigenvectors can be verified directly. The guiding principles of manifest covariance and consistency with QFT led to use of the four-potential in the Lorenz gauge. The photon inner product is then the sum over definite helicity transverse modes where the contributions of scalar and longitudinal photons cancel. All observables are described by Hermitian operators and the position probability amplitude is the projection of the photon’s state onto a basis of position eigenvectors.

We have avoided called $\psi\left(x\right)$ or any of the electromagnetic fields ”the wave function” since singling out any one of these entities seems arbitrary. It was concluded here that all the mathematical apparatus and physical interpretation of classical electromagnetic theory is inherited by this first quantized theory of the photon. On top of this base is added a conserved Lorentz four-current density, an invariant inner product, and Hermitian operators representing observables.

According to the Reeh-Schlieder theorem RS there are no local annihilation or creation operators, so a photon cannot be destroyed locally. In a source free region reality of the field and wave function implies no net absorption. The real photon wave function $\psi_{y\lambda}\left(x\right)=\frac{1}{2}\left[\left(A_{x\lambda},A_{y}\right)+\left(A_{y\lambda},A_{x}\right)\right]$ is an entangled sum of the probability amplitudes for a photon emitted at $y$ and absorbed at $x$ or emitted at $x$ and absorbed at $y$. This is of the form (18) where $x$ and $y$ are on a space-like hyperplane. Individually these terms are nonlocal, only their real sum propagates causally. While the Reeh-Schlieder theorem does not apply directly to a first quantized theory that contains no particle creation or annihilation operators, to provides important physical insight.

With the addition of photon probability density, Maxwell’s theory based on Faraday experiments FaradayMaxwell can be thought of as the first excursion into quantum mechanics. Maxwell’s equations are the original field theory, so it should perhaps not come as a surprise that they have a first quantized interpretation. There is no quantum-classical divide since the quantum theory is just an extension of the classical field theory. Since our formalism is consistent with the quantum electrodynamic creation and annihilation operators, it can be extended to coherent and Fock states. In the absence of sources and sinks there is no net photon absorption since the field is real and positive energy absorption is cancelled out by negative energy emission. If the electric four-current density $J_{e}$ is nonzero, Maxwell’s wave equation is no longer source free and (2) is replaced with $\square A^{\mu}\left(x\right)=-\mu_{0}J_{e}\left(x\right)$ so nonzero net absorption is possible and the spatial integral of the inner product (35) is not conserved.

*Acknowledgement: *I thank Juan Leon for valuable discussion and for convincing us to take Reeh-Schlieder seriously.