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What is a photon?
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According to quantum electrodynamics, the most accurately verified 
theory in physics, a photon is a single-particle excitation of the 
free quantum electromagnetic field. More formally, it is a state of 
the free electromagnetic field which is an eigenstate of the photon 
number operator with eigenvalue 1.

The pure states of the free quantum electromagnetic field
are elements of a Fock space constructed from 1-photon states.
A general n-photon state vector is an arbitrary linear combinations 
of tensor products of n 1-photon state vectors; and a general pure 
state of the free quantum electromagnetic field is a sum of n-photon 
state vectors, one for each n. If only the 0-photon term contributes, 
we have the dark state, usually called the vacuum; if only the 
1-photon term contributes, we have a single photon.

A single photon has the same degrees of freedom as a classical vacuum
radiation field. Its shape is characterized by an arbitrary nonzero
real 4-potential A(x) satisfying the free Maxwell equations, which in 
the Lorentz gauge take the form
   nabla dot nabla A(x) = 0,
   nabla dot A(x) = 0,
expressing the zero mass and the transversality of photons. Thus for
every such A there is a corresponding pure photon state |A>.
Here A(x) is _not_ a field operator but a photon amplitude;
photons whose amplitude differ by an x-independent phase factor are 
the same. For a photon in the normalized state |A>, the observable 
electromagnetic field expectations are given by the usual formulas 
relating the 4-potential and the fields,
   <\E(x)> = <A|\E(x)|A> 
           = - partial \A(x)/partial x_0 - c nabla_\x A_0(x),
and 
   <\B(x)> = <A|\B(x)|A> = nabla_\x x \A(x)
[hmmm. check if the latter really is the case.]
Here \x (fat x) and x_0 are the space part and the time part of a
relativistic 4-vector, \E(x), \B(x) are the electromagnetic
field operators (related to the operator 4-potential by analogous
formulas), and c is the speed of light. Amplitudes A(x) producing 
the same \E(x) and \B(x) are equivalent and related by a gauge 
transformation, and describe the same photon. 


In momentum space (frequently but not always the appropriate choice), 
single photon states have the form 
   |A> = integral d\p^3/p_0 A(\p)|\p>,
where |\p> is a single particle state with definite 3-momentum 
\p (fat p), p_0=|\p| is the corresponding photon energy divided by c,
and the photon amplitide A(\p) is a polarization 4-vector.
Thus a general photon is a superposition of monochromatic waves with 
arbitrary polarizations, frequencies and directions.
(The Fourier transform of A(\p) is the so-called analytic signal
A^(+)(x), and by adding its complex conjugate one gets the real 
4-potential A(x) in the Lorentz gauge.)

The photon amplitude A(\p) can be regarded as the photon's 
wave function in momentum space. Since photons are not localizable
(though they are localizable approximately), there is no 
meaningful photon wave function in coordinate space; see the
next entry in this FAQ. One could regard the 4-potential A(x) as 
coordinate space wave function, but because of its gauge dependence, 
this is not really useful.


This is second quantized notation, as appropriate for quantum fields.
This is how things always look in second quantization, even for a 
harmonic oscillator. The wave function psi(x) or psi(p) in standard 
(first quantized) quantum mechanics becomes the state vector
   psi = integral dx psi(x) |x>   or   integral dp psi(p) |p> 
in Fock space; the wave function at x or p turns into the coefficient 
of |x> or |p>. In quantum field theory, x, A (the photon amplitude), and
E(x) (the electric field operator) correspond to  k (a component of the
momentum), x, and p_k. Thus the coordinate index k is inflated to the 
spacetime position x, the argument of the wave function is inflated to 
a solution of the free Maxwell equations, the momentum operator is 
inflated to a field operator, and the integral over x becomes a 
functional integral over photon amplitudes,
   psi = integral dA psi(A) |A>.
Here psi(A) is the most general state vector in Fock space; for a 
single photon, psi depends linearly on A,
   psi(A) = integral d\p^3/p_0 A(\p)|\p> = |A>.
Observable electromagnetic fields are obtained as expectation values
of the field operators \E(x) and \B(x) constructed by differentiation of
the textbook field operator A(x). As the observed components of 
the mean momentum, say, in ordinary quantum mechanics are
   <p_k> = integral dx psi(x)^* p_k psi(x),
so the observed values of the electromagnetic field are
   <\E(x)> = <psi|\E(x)|psi> = integral dA psi(A)^* \E(x) psi(A).
   <\B(x)> = <psi|\B(x)|psi> = integral dA psi(A)^* \B(x) psi(A).
]
In a frequently used interpretation (valid only approximately),
the term A(\p)|\p> represents the one-photon part of a monochromatic 
beam with frequency nu=cp_0/h, direction \n(\p)=\p/p_0, and 
polarization determined by A(\p). Here h = 2 pi hbar, where hbar is 
Planck's number; omega=cp_0/hbar is the angular frequency.

The polarization 4-vector A(\p) is orthogonal to the 4-momentum p 
composed of p_0 and \p, obtained by a Fourier transform of the 
4-potential A(x) in the Lorentz gauge. (The wave equation translates 
into the condition p_0^2=\p^2, causality requires p_0>0, hence 
p_0=|\p|, and orthogonality p dot A(\p) = 0 expresses the Lorentz 
gauge condition. For massless particles, there remains the additional 
gauge freedom to shift A(\p) by a multiple of the 4-momentum p, which 
can be used to fix A_0=0.) 

A(\p) is usually written (in the gauge with vanishing time component) as
a linear combination of two specific polarization vectors eps^+(p) and 
eps^-(p) for circularly polarized light (corresponding to helicities +1 
and -1), forming together with the direction vector \n(\p) an 
orthonormal basis of complex 3-space. In particular,
   eps^+(p) eps^+(p)^* + eps^-(p)eps^-(p)^* + \n(\p)\n(\p)^* = 1
is the 3x3 identity matrix. (This is used in sums over helicities for 
Feynman rules.) Specifically, eps^+(p) and eps^-(p) can be obtained by 
finding normalized eigenvectors for the eigenvalue problem 
[check. The original eigenvalue problem is p dot J eps = lambda eps.]
   p x eps = lambda eps 
with lambda = +-i|p|. For example, if p is in z-direction then 
   eps^+(p) = (1, -i, 0)/sqrt(2),
   eps^-(p) = (i, -1, 0)/sqrt(2),
and the general case can be obtained by a suitable rotation.
An explicit calculation gives almost everywhere
    eps^+(p) = u(p)/p_0
where p_0=|p| and 
    u_1(p) = p_3 - i p_2 p'/p'',
    u_2(p) = -i p_3 - i p_1 p'/p''
    u_3(p) = p'
with
   p' = p_1+ip_2, 
   p''= p_3+p_0.
[what is eps^-(p)?]
These formulas become singular along the negative p_3-axis,
so several charts are needed to cover 

For experiments one usually uses nearly monochromatic light bundled 
into narrow beams. If one also ignores the directions (which are 
usually fixed by the experimental setting, hence carry no extra
information), then only the helicity degrees of freedom remain,
and the 1-photon part of the beam behaves like a 2-level quantum 
system ('a single spin'). 

A general monochromatic beam with fixed direction in a pure state is 
given by a second-quantized state vector, which is a superposition of 
arbitrary multiphoton states in the Bosonic Fock space generated by 
the two helicity degrees of freedom. This is the basis for most 
quantum optics experiments probing the foundations of quantum 
mechanics. 


The simplest state of light (generated for example by 
lasers) is a coherent state, with state vector proportional to
    e(A) = |vac> + |A> + 1/sqrt(2!) |A> tensor |A> 
                 + 1/sqrt(3!) |A> tensor |A> tensor |A> + ...
where |A> is a one-photon state. Thus coherent states also have the 
same degrees of freedom as classical electromagnetic radiation. 
Indeed, light in coherent states behaves classically in most respects.

At low intensity, the higher order terms in the expansion are
negligible, and since the vacuum part is not directly observable,
a low intensity coherent states resembles a single photon state.

On the other hand, ordinary light is essentially never, and high-tech 
light almost never, describable by single photons. True single photon 
states are very hard to produce to good accuracy, and were created 
experimentally only recently:
   B.T.H. Varcoe, S. Brattke, M. Weidinger and H. Walther,
   Preparing pure photon number states of the radiation field, 
   Nature 403, 743--746 (2000).
see also 
   http://www.qis.ucalgary.ca/quantech/fock.html 


A good informal discussion of what a photon is from a more practical 
perspective was given by Paul Kinsler in
   http://www.lns.cornell.edu/spr/2000-02/msg0022377.html
But this does not tell the whole story. An interesting collection of 
articles explaining different current views is in 
    The Nature of Light: What Is a Photon?
    Optics and Photonics News, October 2003
    http://www.osa-opn.org/Content/ViewFile.aspx?Id=3185
Further discussion is given in the section ''Coherent states of light 
as ensembles'' of the present FAQ, and in my slides
    http://www.mat.univie.ac.at/~neum/ms/lightslides.pdf
    http://www.mat.univie.ac.at/~neum/ms/optslides.pdf

The standard reference for quantum optics is
   L. Mandel and E. Wolf,
   Optical Coherence and Quantum Optics, 
   Cambridge University Press, 1995.
Mandel and Wolf write (in the context of localizing photons),
about the temptation to associate with the clicks of a photodetector
a concept of photon particles. [If there is interest, I can try to 
recover the details.] The wording suggests that one should resist the
temptation, although this advice is usually not heeded. However,
the advice is sound since a photodetector clicks even when it
detects only classical light! This follows from the standard analysis
of a photodetector, which treats the light classically and only 
quantizes the detector. Thus the clicks are an artifact of 
photodetection caused by the quantum nature of matter, rather than
a proof of photons arriving!!!


A coherent light source (laser) produces a coherent state of light, 
which is a superposition of the vacuum state, a 1-photon state,
a 2-photon state, etc, with squared amplitudes given by a Poisson 
distribution. At low intensity, this is misinterpreted in practice 
as random single photons arriving at the end of the beam in a 
random Poisson process, because the photodetector produces clicks 
according to this distribution. 

Incoherent light sources usually consist of thermal mixtures and 
produce other distributions, but otherwise the description (and 
misinterpretation) is the same. 
Nevertheless, one must understand this misinterpretation in order
to follow much of the literature on quantum optics.

Thus the talk about photons is usually done inconsistently;
almost everything said in the literature about photons should be taken 
with a grain of salt. 
There are even people like the Nobel prize winner Willis E. Lamb 
(the discoverer of the Lamb shift) who maintain that photons don't 
exist. See towards the end of
   http://web.archive.org/web/20040203032630/www.aro.army.mil/phys/proceed.htm
The reference mentioned there at the end appeared as 
   W.E Lamb, Jr., 
   Anti-Photon, 
   Applied Physics B 60 (1995), 77--84
This, together with the other reference mentioned by Lamb, is reprinted 
in 
   W.E Lamb, Jr., 
   The interpretation of quantum mechanics, 
   Rinton Press, Princeton 2001.

I think the most apt interpretation of an 'observed' photon as used
in practice (in contrast to the photon formally defined as above) is 
as a low intensity coherent state, cut arbitrarily into time slices 
carrying an energy of h*nu = hbar*omega, the energy of a photon at 
frequency nu and angular frequency omega. 
Such a state consists mostly of the vacuum (which is not directly 
observable hence can usually be neglected), and the contributions of 
the multiphoton states are negligible compared to the single photon 
contribution. 
With such a notion of photon, most of the actual experiments done make 
sense, though it does not explain the quantum randomness of the 
detection process (which comes from the quantized electrons in the 
detector).


A nonclassical description of the electromagnetic field where states of 
light other than coherent states are required is necessary mainly for
special experiments involving recombining split beams, squeezed
state amplification, parametric down-conversion, and similar 
arrangements where entangled photons make their appearance.
There is a nice booklet on this kind of optics:
   U. Leonhardt,
   Measuring the Quantum State of Light,
   Cambridge, 1997.

Nonclassical electromagnetic fields are also relevant in the
scattering of light, where there are quantum corrections 
due to multiphoton scattering. These give rise to important effects 
such as the Lamb shift, which very accurately confirm the quantum
nature of the electromagnetic field. They involve no observable 
photon states, but only virtual photon states, hence they are unrelated 
to experiments involving photons. Indeed, there is no way to observe 
virtual particles, and their name was chosen to reflect this. 
(Observed particles are always onshell, hence massless for photons, 
whereas it is an easy exercise that the virtual photon mediating 
electromagnetic interaction of two electrons in the tree approximation 
is never onshell.)