Abstract
We provide a suitable theoretical foundation for the notion of the quantum coherent state which describes the electrostatic field due to a static external macroscopic charge distribution introduced by the author in 1998 and use it to rederive the formulae obtained in 1998 for the inner product of a pair of such states. (We also correct an incorrect factor of \(4\pi\) in some of those formulae.) Contrary to what one might expect, this inner product is usually non-zero whenever the total charges of the two charge distributions are equal, even if the charge distributions themselves are different. We actually display two different frameworks that lead to the same inner-product formulae, in the second of which Gauss’s law only holds in expectation value. We propose an experiment capable of ruling out the latter framework. We then address the problem of finding a product picture for QED—i.e. a reformulation in which it has a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a Hilbert space which is a subspace (the physical subspace) of the full tensor product of a charged-matter Hilbert space and an electromagnetic-field Hilbert space. (The traditional Coulomb gauge formulation of QED isn’t a product picture in this sense because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) Motivated by the first framework for our coherent-state construction, we find such a product picture and exhibit its equivalence with Coulomb gauge QED both for a charged Dirac field and also for a system of non-relativistic charged balls. For each of these systems, in all states in the physical subspace (including the vacuum in the case of the Dirac field) the charged matter is entangled with longitudinal photons and Gauss’s law holds as an operator equation; albeit the electric field operator (and therefore also the full Hamiltonian) while self-adjoint on the physical subspace, fails to be self-adjoint on the full tensor-product Hilbert space. The inner products of our electrostatic coherent states and the product picture for QED are relevant as analogues to quantities that play a rôle in the author’s matter-gravity entanglement hypothesis. Also, the product picture provides a temporal gauge quantization of QED which appears to be free from the difficulties which plagued previous approaches to temporal-gauge quantization.
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Notes
We explicitly include \(\epsilon _0\) and \(\hbar\) and c in our opening paragraphs but later set them all to 1. We are of course free to choose \(\epsilon _0\) to be whatever we like (provided the permeability of vacuum, \(\mu _0\) is taken to be \(1/\epsilon _0 c^2\)) but whatever we choose of course affects our unit of charge. We caution the reader that, in [7], \(\epsilon _0\) was taken to be \(1/4\pi\). Let us also note here that we may restore \(\epsilon _0\), \(\hbar\) and c in all our later equations by noting that \(\pi = -\epsilon _0 E\) and by making the following insertions in our equations: (a) a factor of \(\epsilon _0^{-1}\) in front of the term \(\frac{1}{2}{\varvec{\pi }}^2\) and a factor of \(\epsilon _0 c^2\) (\(=1/\mu _0\)) in front of the term \(\frac{1}{2}({\varvec{\nabla }}{\varvec{\times }}\varvec{A})^2\) in the Hamiltonian [(13) and subsequent equations] for the free electromagnetic field; (b) a factor of c in front of each factor of k in Equation (26) which relates the \(\phi\) and \(\pi\) of a scalar field to creation and annihilation operators and a factor of \(\epsilon _0 c\) in front of each factor of k in all equations [(36), (47) etc.] which relate creation and annihilation operators to electromagnetic \({\varvec{\pi }}\) and \(\varvec{A}\) etc.; (c) [in consequence of (b)] the \(\chi (\varvec{k})\) of Equation (28) becomes \(|ck|^{1/2}\phi _{\text {sc}}(\varvec{k})/\sqrt{2}\), while the \(\chi ^i(\varvec{k})\) of (60) becomes \(k^i\phi ({\varvec{k}})/\sqrt{2\epsilon _0 c}|k|^{1/2}\); (d) a factor of \(\hbar\) on the right hand side of all commutation and anticommutation relations; (e) a factor of \(1/\hbar\) in the (complex) exponents in all expressions (24), (50), (85) etc. for the operator(s) we call U and other related exponentials as well as a factor of \(1/\hbar\) in the exponents in Equations (29), (30) etc. and in front of the \(D_0\) of (32) and similarly for Eq. (61) etc. and in front of the \(D_1\) of (62), as well, of course, as in the exponent of any expression of form \(\exp (-iHt)\) where H is a Hamiltonian; (f) a factor of \(\hbar\) in front of \(\varvec{k}{\varvec{\cdot }} \varvec{a}\) in Eq. (118) and subsequent related equations including in Footnote 7; (g) A factor of c in front of the free electromagnetic Hamiltonian when expressed in terms of creation and annihilation operators—i.e. in Eq. (114) and (115).
Note that with these conventions, we have \([a(\varvec{x}), a^+(\varvec{x}')] = \hbar \delta ^{(3)}(\varvec{x} - \varvec{x}')\), \([a(\varvec{k}), a^+(\varvec{k}')] = \hbar \delta ^{(3)}(\varvec{k} + \varvec{k}')\) etc.
We write “usually” because we are unaware of any example for which we have two different charge distributions whose total charges are the same, for which the two corresponding electrostatic coherent states are orthogonal, except for the simple capacitor example of Sect. 2.5 (where the total charge of the two distributions is zero). And note that that example is on a (flat) space with a different global topology (\({\mathbb {R}}\) times the 2-torus) from \({\mathbb {R}}^3\). See Footnote 8.
One way to convince oneself that the asymptotic formula (9), and also the formula (120) of Sect. 4.3, should hold for charged bodies with a wide range of shapes and charge distributions—and also to check the correctness of the numerical factor \(1/4\pi ^2\) in (9) and (120)—is as follows: First notice that the left hand side of (120) (of which (62) [re-expressed as in Eq. (31) of Footnote 7] is the special case where \(q_1=q_2\)) is, by (60), given by the momentum-space integral
$$ \int {\frac{k}{2}} \tilde{\phi }_{1} \left( \varvec{k} \right)*\left( {{1 - e^{{i\varvec{k} \cdot \varvec{a}}}} } \right)\tilde{\phi }_{2} \left(\varvec{k} \right)d^{3} k $$(10)where \({{\tilde{\phi }}}_i(\varvec{k})\), \(i=1,2\) denote the Fourier transforms (see Footnote 6) of \(\phi _i(\varvec{x})\), and \(\phi _1({\varvec{x}})\) and \(\phi _2(\varvec{x})\) denote the electrical potentials of the charged objects when their centres are located (say) at the origin. (Here we just suppose we have defined some suitable notion of ‘centre’ for each of the charged matter distributions involved in (120).) Then notice that, in the case of pointlike charges for which \(\rho _i(\varvec{x}) = q_i\delta ^{(3)}(\varvec{x})\), \(i = 1,2\), by (3) (with \(\epsilon _0=1\)) and Fourier transformation, we have \({{\tilde{\phi }}}_i(\varvec{k}) = q_i/(2\pi )^{3/2}k^2\). So, formally, (10) becomes
$$ \begin{gathered} \frac{{q_{i} q_{j} }}{{16\pi ^{2} }}\int {\frac{1}{{k^{3} }}} \left( {1 - e^{{i\varvec{k} \cdot \varvec{a}}} } \right){\mkern 1mu} d^{3} k, \hfill \\ \quad = \frac{{q_{i} q_{j} }}{{4\pi }}\int\limits_{0}^{\infty } {\frac{1}{k}} \left( {1 - \frac{{\sin \left( {ka} \right)}}{{ka}}} \right)dk. \hfill \\ \end{gathered} $$(11)The integral in (11) is of course divergent. However, its formal derivative with respect to a,
$$\begin{aligned} \int _0^\infty \frac{\sin (ka) - ka\cos (ka)}{k^2a^2}\, dk, \end{aligned}$$is convergent and (in view of the fact that \(\int _0^\infty \frac{\sin \kappa - \kappa \cos \kappa }{\kappa ^2}\, d\kappa = [-\sin \kappa /\kappa ]_0^\infty = 1\)) is equal to 1/a. Hence it is reasonable to assign the value \(\ln (a/R)\) to the integral in (11), where R is an unfixable constant. This further strongly suggests that when one replaces point charges by our charge distributions, the same formula will hold asymptotically for large a but now that, for any given (pair of) charge distribution(s), R will be fixed.
See also [1] for a different but relevant consideration written with the linearized gravity case in mind but equally relevant for quantum electrostatics.
As far as we can see now, the calculations in [7] of \(D_0\) for the ‘spin-zero gravity’ model there (which is the same as the scalar model discussed in Sect. 2.2 here) were carried out correctly. Also the statement in [7]) to the effect that (as we put it in Sect. 2.4 here) the electrostatic (or ‘spin-1’) decoherence exponent, \(D_1\), is equal to the spin-0 exponent, \(D_0\), when the classical static scalar charge densities \(\sigma _1\) and \(\sigma _2\), are equated with \(\rho _1\) and \(\rho _2\), is correct. However, unfortunately, the formulae given in [7] for \(D_1\) were a factor of \(4\pi\) bigger than the correct formulae (which are given here). Let us also note that the argument and claim in [7] that (in the sense explained there) the spin-2 decoherence exponent, \(D_2\) is a factor of 6 times bigger than \(D_0\) also appears to be in error. This error also infects [1, 15]. It is intended to correct it in the forthcoming paper [11].
As was pointed out in [7], there is a mathematical subtlety here due to the fact that our quantum scalar field is massless. Due to the 1/x tail in \(\phi _{\text {sc}}\), the putative one-particle Hilbert space vector \(\chi\) of (28) will not be normalizable and so does not strictly belong to the one-particle Hilbert space and so \(e^{-i\pi (\phi _{\text {sc}})}\) is not strictly a unitary operator (the integrand in the integral \(\int \pi ({\varvec{x}})\phi _{\text {sc}}(\varvec{x}) d^3x\) has an infra-red [i.e. large distance] divergence in it) and the coherent state \(\varPsi\) of (24) does not strictly belong to the augmented Fock space. In terms of notions from the algebraic approach to quantum field theory [5] it should really be understood as a state (namely the composition of the vacuum state with the automorphism \(\varphi \mapsto \varphi + \phi _{\text {sc}}\)—see (34)) in the sense of a positive linear functional on the relevant field algebra, and a change of Hilbert-space representation should be invoked. However, we will proceed as if these vectors were normalizable since, as long as the sources, \(\sigma\), of the scalar field, \(\varphi\), that they describe have the same integral \(\int \sigma (\varvec{x})\, d^3x\) (analogous to the total charge in the electrostatic case) the quantity we wish to compute, i.e. \(|\langle \varPsi _1|\varPsi _2\rangle |\), given by the formula (32), will be finite. This is because, as long as those integrals are equal, the infra-red divergences in each of the terms, \(\chi _1\), \(\chi _2\) in (32) will cancel out. We will assume that finite result to be the physically correct value and we expect that this can be demonstrated rigorously.
Similar remarks apply to the electrostatic coherent state (53) and in the linearized gravity case discussed in [7] and to be discussed further in [11]. Also the integrands in the exponents in the definitions of the formally unitary operators, U of (50), U of (85) in Sect. 3 and its counterpart U in Sect. 4 all have infra-red divergencies similar to that in \(\int \pi ({\varvec{x}})\phi _{\text {sc}}(\varvec{x}) d^3x\) while the U of (85) in Sect. 3 additionally has ultra-violet divergencies both from the short distance 1/r divergence and and also to the field product \(\psi ^*(x)\psi (x)\) in the \(\phi\) of (82). Thus the physical subspaces of Sects. 3 and 4 are not strictly subspaces of the relevant QED augmented Hilbert spaces. However, just as we obtained finite results for the inner products of coherent states in Sect. 2, we obtain finite results e.g. for the reduced density operator of Sect. 4.3.
Throughout the paper, we adopt the convention that the Fourier transform \({{\tilde{f}}}(\varvec{k})\) of a function \(f(\varvec{x})\) on \({\mathbb R}^3\) is \((2\pi )^{-3/2}\int f(\varvec{x}) e^{-i\varvec{k}{\varvec{\cdot }}\varvec{x}}\, d^3 x\). We often omit the tilde if it is clear from the context (or from the fact that the value is denoted \(\varvec{k}\) rather than \(\varvec{x}\)) that it is the Fourier transform that is being referred to.
In this footnote, we give some more details on the derivation of Eqs. (29) and (32) and also Eqs. (61) and (62). The derivation (30) of Eq. (29) makes use of the special case, \(e^Be^A=e^{A+B-[A,B]/2}\) of the Baker-Campbell-Hausdorff (BCH) relation for a pair of operators when their commutator is a c-number (applied to \(a(\psi )\) and \(a^+(\phi )\)).
As for Eqs. (32), it is easy to see from the commutation relations (27) for our creation and annihilation operators, using the same BCH formula, or rather \(e^Be^A = e^{-[A,B]} e^Ae^B\), that if \(\varPsi _1\) and \(\varPsi _2\) are as in (29) (with \(\chi\) replaced by \(\chi _1\) and \(\chi _2\) respectively) then \(\langle \varPsi _1|\varPsi _2\rangle = \exp (-\langle \chi _1|\chi _1\rangle /2 - \langle \chi _2|\chi _2\rangle /2+\langle \chi _1|\chi _2\rangle )\). So \(|\langle \varPsi _1|\varPsi _2\rangle | = \exp (-\Vert \chi _1-\chi _2\Vert ^2/2)\).
If \(\chi _1\) and \(\chi _2\) are related by a spatial translation (so we would also say that \(\varPsi _1\) and \(\varPsi _2\) are related by that spacelike translation) then, in momentum space, we will have, say \(\chi _2(\varvec{k}) = \exp (i\varvec{k}{\varvec{\cdot }}\varvec{a})\chi _1(\varvec{k})\) and hence one sees, by thinking of it as an integral in momentum space, that \(\langle \chi _1|\chi _2\rangle\) is real and hence \(\langle \varPsi _1|\varPsi _2\rangle = |\langle \varPsi _1|\varPsi _2\rangle |\). Writing the latter as \(\exp (-D_0)\), we thus confirm (32). We also easily conclude that \(D_0\) has the following alternative forms:
$$\begin{aligned} D_0=\Vert \chi _1-\chi _2\Vert ^2/2 = \langle \chi |(1-\cos (\varvec{k}{\varvec{\cdot }}\varvec{a}))\chi \rangle =\langle \chi |(1-\exp (i(\varvec{k}{\varvec{\cdot }}\varvec{a})))\chi \rangle . \end{aligned}$$(31)Similar results to those discussed above hold for electrostatic coherent states and \(D_1\). In particular, to derive (61), we first note (cf. the paragraph ending with Eq. (30)) that the \({\varvec{\chi }}(\varvec{k})\) of (60) is an odd function of \(\varvec{k}\) (unlike the \(\chi ({\varvec{k}})\) of (28) which is even) and thus
$$\begin{aligned} {[}a_i(\chi ^i), a^+_j(\chi ^j)] = \int \chi ^i(-\varvec{k})\chi ^i(\varvec{k})\, d^3k = - \int {\chi ^*}^i(\varvec{k})\chi ^i(\varvec{k})\, d^3k = -\langle \chi ^i|\chi ^i\rangle . \end{aligned}$$We then have (cf. (30))
$$\begin{aligned} e^{i\int {{\hat{A}}}^i\partial _i\phi \, d^3x}\varOmega= & {} e^{-a^+_i(\chi ^i)-a_i(\chi ^i)}\varOmega = e^{[a_i(\chi ^i), a^+_j(\chi ^j)]/2}e^{-a^+_i(\chi ^i)}e^{-a_i(\chi ^i)}\varOmega \\= & {} e^{[a_i(\chi ^i), a^+_j(\chi ^j)]/2}e^{-a^+_i(\chi ^i)}\varOmega = e^{-\langle \chi ^i|\chi ^i\rangle /2}e^{-a^+_i(\chi ^i)}\varOmega . \end{aligned}$$Finally, the derivation of (62) is similar to that of (32) indicated above. (Essentially all of the results referred to here were already stated in [7].)
The vanishing of \(|\langle \varOmega |\varPsi \rangle |\) in the simple capacitor model with two plates and periodic boundary conditions is due to an infra-red divergence in the counterpart \(\left( Q^2\int _{-\infty }^\infty \frac{\sin ^2(ka/2)}{k^3}\, dk/2\pi L^2\right)\) for the simple capacitor to the formula \(2Q^2\int _{-\infty }^\infty \frac{\sin ^4(ka/2)}{k^3}\, dk/\pi L^2\) (see after Eq. (69)) for the double capacitor that can be traced to the fact that, unlike in the double capacitor, where the classical electrical potential can be taken to vanish above the top plate and below the bottom plate (one could earth them both!—see also Fig. 2) in the simple capacitor with those periodic boundary conditions, the classical electrostatic potential must be a non-zero constant either above the top plate or below the bottom plate. (One can only earth one of the plates!) This infra red divergence is expected to be an artefact of our periodic boundary conditions since the potential will vanish at infinity for finite physical plates with edges. Note, though, that we expect that it can be taken as a signal that, in the absence of periodic boundary conditions, edge-effect corrections to \(|\langle \varOmega |\varPsi \rangle |\) will be more severe for a simple capacitor than for a double capacitor. But we haven’t investigated this.
When we say that \(\tilde{{\varvec{\pi }}}^\perp\) acts non-trivially only on the \({{{\mathcal {F}}}}(\mathcal H_{\text {one}}^{\text {trans}})\) component of \({{{\mathcal {F}}}}(\mathcal H_{\text {one}})\otimes {{{\mathcal {H}}}}_{\text {Dirac}}\), and that \(\phi\) acts non-trivially only on the \({{{\mathcal {H}}}}_{\text {Dirac}}\) component of \({{{\mathcal {F}}}}({{\mathcal {H}}}_{\text {one}})\otimes \mathcal{H}_{\text {Dirac}}\), we mean that, when the QED augmented Hilbert space, \({{{\mathcal {F}}}}({{\mathcal {H}}}_{\text {one}})\otimes \mathcal{H}_{\text {Dirac}}\), is written as \({{\mathcal {F}}}(\mathcal H_{\text {one}}^{\text {trans}})\otimes {{{\mathcal {F}}}}(\mathcal H_{\text {one}}^{\text {long}})\otimes {{{\mathcal {H}}}}_{\text {Dirac}}\), then \(\tilde{{\varvec{\pi }}}^\perp\) takes the form \(\tilde{{\varvec{\varpi }}}^{\text {trans}}\otimes \text {id}\otimes \text {id}\) for some operator \(\tilde{{\varvec{\varpi }}}^\perp\) on \({{\mathcal {F}}}(\mathcal H_{\text {one}}^{\text {trans}})\) and that \(\phi\) takes the form \(\text {id}\otimes \text {id}\otimes \varphi\) for some operator \(\varphi\) on \({{{\mathcal {H}}}}_{\text {Dirac}}\) where \(\text {id}\) denote identity operators.
The reason we model our particles as having extended charge distributions can be traced to the fact that, while the inner product between logitudinal coherent photon states for the Coulomb potential of two different point charges in different locations is formally finite, the inner product of such a state for a given single point charge with itself (i.e. the square of that state’s norm) is infinite. The latter infinity is reflected in the nonexistence of the limit \(R\rightarrow 0\) in (6), (7), (9) and in the appearance of the denominator R in the terms in the first product in (121), while the former finiteness is reflected in the fact that the factors of R cancel out in the remaining terms of that product. This infinity is of course related to the fact that the Coulomb potential of a pointlike charged particle due to itself (i.e. the ‘self potential’ of a point charge) is infinite. In Coulomb gauge, this doesn’t prevent one from having a model with pointlike charged particles. One simply omits the self-energy terms i.e. the terms with \(I=J\) in the formula (99) for \(V_{\text {Coulomb}}\). The problem is that, in order to transform a Coulomb gauge Hamiltonian to the product picture, we need the formula for \(V_{\text {Coulomb}}\) to arise as \(\frac{1}{2}\int {\varvec{\nabla }}\phi {{\varvec{\cdot }}}{\varvec{\nabla }}\phi \, d^3x\) for some potential function \(\phi\). But the point-particle \(V_{\text {Coulomb}}\) of (99) where the terms with \(I =J\) are necessarily omitted from the sum (because they are infinite!) cannot be written in this way. (Instead, if \(\phi\) denotes the sum \(\sum _{I=1}^N\phi _I\) where \(\phi _I\) is the potential of the Ith ball, then the latter point-particle \(V_{\text {Coulomb}}\) arises as the limit as the charge densities approach delta functions of \(\int {\varvec{\nabla }}\phi {{\varvec{\cdot }}}{\varvec{\nabla }}\phi \, d^3x - \sum _{I=1}^M \int {\varvec{\nabla }}\phi _I{{\varvec{\cdot }}}{\varvec{\nabla }}\phi _I\, d^3x\).) Indeed, it might well be negative, as, e.g. in the case where \(N=2\) and \(q_1\) and \(q_2\) have opposite signs. See, in this connection the remark that “negative inter-ball potentials \(\dots\) are possible \(\dots\)” towards the end of Sect. 4.2. See also the further discussion below.
The word ‘epiphenomenon’ is used to mean several different things. So let us clarify here that, when we say, in Sect. 1.2, that, in Coulomb gauge, the longitudinal modes of the electric field are an epiphenomenon of the charged matter, we are referring to the fact that the equation \(\varvec{E}^{\text {long}}= -{\varvec{\nabla }}\phi\) is, in a Coulomb gauge understanding, regarded as a definition of the left hand side in terms of the right hand side, and not an equation between two independently defined things. As far as I understand, this use of the word is similar to the use of the word when one says, e.g., that the mind may be an epiphenomenon of brain activity.
Indeed, an obvious place where our analogy in Sect. 5 might be relevant is the mind-body problem, where there is a well known and long standing controversy as to whether, on the one hand, the mind is either an epiphenomenon of brain activity, or in a more radical variation, does not exist at all, or, on the other hand, whether it is a real physical thing in its own right. Pursuing the analogy, one is tempted to speculate that perhaps it is possible to have a theory of the mind in which one has a choice as to whether to regard it as an epiphenomenon of brain activity or even to regard it as not existing at all, and another theory in which it is a real physical thing in its own right, and yet the two theories turn out, similarly to in our QED analogy, to be operationally indistinguishable. And, if we pursue the analogy further, we might expect that, were one were to adopt the theory in which it is a real physical thing, then new questions might arise and be answerable—just as new questions arise and can be answered in our product picture (such as “What is the partial trace of a total state of QED over the electromagnetic field”).
Of course, there are several other places in physics that provide such analogies. Indeed, classical electrodynamics does. lts Hamiltonian formulation in Coulomb gauge goes together with the ontology that the longitudinal modes of the electric field are either an epiphenomen of the charged matter variables, or don’t exist at all; while, on the other hand the classical Maxwell’s equations go together with the ontology that all modes of the electromagnetic field are real. However, having the product picture in QED would be like having an alternative Hamiltonian formulation of the classical Maxwell equations, and as far as I know, there is no such thing in classical electrodynamics. Also the notions of partial trace and entanglement don’t have classical counterparts and these perhaps provide a particularly clear example of how new questions can arise in what, from another point of view, is an equivalent theory.
Finally, since Maxwell himself said that ‘charge’ and ‘current’ are ‘epiphenomena’ of electromagnetic fields, we should clarify that what he had in mind in saying that appears to be unrelated to the sorts of issues and analogies on which we are focusing here.
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Acknowledgements
The author thanks the Leverhulme Foundation for the award of Leverhulme Fellowship RF&G/9/RFG/2002/0377 for the period October 2002 to June 2003 during which some of this work was done. I thank Michael Kay for valuable comments.
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Kay, B.S. Quantum Electrostatics, Gauss’s Law, and a Product Picture for Quantum Electrodynamics; or, the Temporal Gauge Revised. Found Phys 52, 6 (2022). https://doi.org/10.1007/s10701-021-00512-2
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DOI: https://doi.org/10.1007/s10701-021-00512-2