A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.

1.
I.
Bialynicki-Birula
, in
Progress in Optics
, edited by
E.
Wolf
(
Elsevier
,
Amsterdam
,
1996
), Vol. XXXVI, p.
245
.
2.
A. I.
Akhiezer
and
V. B.
Berestetskii
,
Quantum Electrodynamics
(
Interscience
,
New York
,
1965
).
3.
L.
Landau
and
R.
Peierls
,
Z. Phys.
62
,
188
(
1930
).
4.
W.
Pauli
,
General Principles of Quantum Mechanics
(
Springer
,
Berlin
,
1980
).
5.
J. F.
Sipe
,
Phys. Rev. A
52
,
1875
(
1995
).
6.
H.
Weber
,
Die Partiellen Differential-Gleichungen der Mathematischen Physik nach Riemann’s Vorlesungen
(
Friedrich Vieweg und Sohn
,
Braunschweig
,
1901
), p.
348
;
L.
Silberstein
,
Ann. Phys.
327
,
579
(
1907
);
L.
Silberstein
,
Ann. Phys.
329
,
783
(
1907
);
L.
Silberstein
The Theory of Relativity
(
MacMillan
,
London
,
1914
).
7.
R. H.
Good
,Jr.
,
Phys. Rev.
105
,
1914
(
1957
);
I.
Bialynicki-Birula
,
Acta Phys. Pol., A
86
,
97
(
1994
);
I.
Bialynicki-Birula
,
Phys. Rev. Lett.
80
,
5247
(
1998
);
I.
Bialynicki-Birula
, ,
Coherence and Quantum Optics VII
, edited by
J. H.
Eberly
,
L.
Mandel
, and
E.
Wolf
(
Plenum
,
New York
,
1996
), p.
313
.
8.
T. D.
Newton
and
E. P.
Wigner
,
Rev. Mod. Phys.
21
,
400
(
1949
).
9.
A. S.
Wightman
,
Rev. Mod. Phys.
34
,
845
(
1962
).
10.
D.
Rosewarne
and
S.
Sakar
,
Quantum Opt.
4
,
405
(
1992
).
11.
J. M.
Jauch
and
C.
Piron
,
Helv. Phys. Acta
40
,
559
(
1967
), http://www.e-periodica.ch/digbib/doasearch;jsessionid=821A9B1D91AFEA7DA2A2869040543A7C.
12.
W. O.
Amrein
,
Helv. Phys. Acta
42
,
149
(
1969
).
13.
E. R.
Pike
and
S.
Sarkar
,
Phys. Rev. A
35
,
926
(
1987
).
14.
M.
Hawton
and
W. E.
Baylis
,
Phys. Rev. A
64
,
012101
(
2001
).
15.
M. H. L.
Pryce
,
Proc. R. Soc. A
195
,
62
(
1948
).
16.
T. F.
Jordan
and
N.
Mukunda
,
Phys. Rev.
132
,
1842
(
1963
);
R. A.
Berg
,
J. Math. Phys.
6
,
34
(
1965
);
T. F.
Jordan
,
J. Math. Phys.
21
,
2028
(
1980
);
J.
Mourad
,
Phys. Lett. A
182
,
319
(
1993
);
H.
Bacry
,
Localizability and Space in Quantum Physics
(
Springer
,
Berlin
,
1988
).
17.
M.
Hawton
,
Phys. Rev. A
59
,
954
(
1999
).
18.
M.
Hawton
and
W. E.
Baylis
,
Phys. Rev. A
71
,
033816
(
2005
).
19.
M.
Hawton
,
Phys. Rev. A
75
,
062107
(
2007
);
M.
Hawton
, “
The Nature of Light: What are Photons?
,”
Proc. SPIE
6664
,
666408
(
2007
);
preprint arXiv:0711.0112;
M.
Hawton
and
V.
Debierre
, preprint arXiv:1512.06067.
20.
M.
Hawton
,
Phys. Rev. A
59
,
3223
(
1999
).
21.
A.
Mostafazadeh
,
Classical Quantum Gravity
20
,
155
(
2003
).
22.
A.
Mostafazadeh
,
Ann. Phys.
309
,
1
(
2004
).
23.
A.
Mostafazadeh
,
Int. J. Mod. Phys. A
21
,
2553
(
2006
).
24.
A.
Mostafazadeh
and
F.
Zamani
,
Ann. Phys.
321
,
2183
(
2006
).
25.
F.
Zamani
and
A.
Mostafazadeh
,
J. Math. Phys.
50
,
052302
(
2009
).
26.
A.
Mostafazadeh
,
Phys. Scr.
82
,
038110
(
2010
).
27.

We use the term Hermitian and self-adjoint synonymously. Note that the Hermiticity of observables is an absolutely inescapable consequence of the condition that the expectation values of observables must be real.32 A non-Hermitian operator can have a real spectrum, but there will always be states in which the expectation value of this operator is not real.

28.

Often the difficulties associated with infinite dimensional Hilbert spaces have to do with the fact that relevant operators are only defined on a proper dense subset of this space. The history of quantum mechanics teaches us that these difficulties are purely mathematical in nature. Indeed the physical state vectors that one can prepare for a measurement do not fill the whole Hilbert space.

29.

A linear operator L:XY defined on X is called an isometry if under its action the inner product of elements of X do not change. An onto isometry is called a unitary operator.

30.

In Dirac’s bra-ket notation, the symbol y| is used for δy.

31.
A.
Mostafazadeh
,
J. Math. Phys.
43
,
205
(
2002
);
A.
Mostafazadeh
,
J. Math. Phys.
43
,
2814
(
2002
).
32.
A.
Mostafazadeh
,
Int. J. Geom. Methods Mod. Phys.
7
,
1191
(
2010
); e-print arXiv:0810.5643.
33.
A.
Mostafazadeh
,
J. Math. Phys.
44
,
974
(
2003
).
34.

The inner products (46) were originally obtained in Ref. 21 using various properties of pseudo-Hermitian operators. They admit a manifestly Lorentz-invariant expression involving a conserved complex current density. See Ref. 24 for details.

35.
I.
Bialynicki-Birula
,
Phys. Rev. Lett.
80
,
5247
(
1998
).
36.
G. C.
Hegerfeldt
,
Phys. Rev. D
10
,
3320
(
1974
).
37.
N.
Barat
and
J. C.
Kimball
,
Phys. Lett. A
308
,
110
(
2003
).
38.

The position operators map scalar fields to scalar fields, so they do not change under Lorentz transformations. The position wave function and the probability density are determined by the inner product of pairs of scalar fields which is also Lorentz-invariant.

39.
D. G.
Currie
,
T. F.
Jordan
, and
E. C. G.
Sudarshan
,
Rev. Mod. Phys.
35
,
350
(
1963
).
40.
J. D.
Jackson
,
Classical Electrodynamics
(
Wiley
,
New York
,
1975
).
41.

Because iD1/20 is Lorentz-invariant,24 Ac also corresponds to a photon field.

42.

These generate a spin-1 representation of the rotation group.

43.

This means that this operator maps elements of H to elements of H.

44.
I. S.
Gradshteyn
and
I. M.
Ryzhik
,
Table of Integrals, Series, and Products
, 7th ed. (
Academic Press
,
Burlington, Massachusetts
,
2007
).
45.

Ref. 14 offers an interesting geometric interpretation of the general helicity eigenvectors in terms of Euler rotations.

46.

Here and in what follows, the powers of D is defined in terms of its spectral representation, i.e., (Dαϕ)(x):=(2π)3/2R3d3k(k2+m2)αeikxϕ̃(k), where ϕ̃(k):=(2π)3/2R3d3xeikxϕ(x) is the Fourier transform of ϕ(x). This in turn implies that (Dαϕ)(x)=R3d3yKα(xy)ϕ(y), where Kα(x):=(2π)3R3d3k(k2+m2)αeikx.

47.

The standard textbook proof of the identification of the probability density (of nonrelativistic QM) with the time component of a four-vector applies only for Hamiltonians that are quadratic polynomials in momenta. For example, it fails for a Hamiltonian of the form H = p4 + v(x), whose use is not prohibited by any of the axioms of quantum mechanics.

You do not currently have access to this content.