Particle positions and the position operator
The standard probability interpretation for quantum particles
is based on the Schr"odinger wave function psi(x), a square integrable
single- or multicomponent function of position x in R^3.
Indeed, with ^* denoting the conjugate transpose,
rho(x) := psi(x)^*psi(x)
is generally interpreted as the probability density to find (upon
measurement) the particle at position x. Consequently,
Pr(Z) := integral_Z dx |psi(x)|^2
is interpreted as the probability of the particle being in the open
subset Z of position space. Particles in highly localized states
are then given by wave packets which have no appreciable size
|psi(x)| outside some tiny region Z.
If the position representation in the Schr"odinger picture exists,
there is also a vector-valued position operator x, whose components
act on psi(x) by multiplication with x_j (j=1,2,3). In particular,
the components of x commute, satisfy canonical commutation relations
with the conjugate momentum
p = -i hbar partial_x,
and transform under rotations like a 3-vector, so that the commutation
relations with the angular momentum J take the form
[J_j,x_k] = i eps_{jkl} x_l.
Moreover, in terms of the (unnormalizable) eigenstates |x,m> of the
position operator correponding to the spectral value x (and a label m
to distinguish multiple eigenstates) we can recover the position
representation from an arbitrary representation by defining psi(x)
to be the vector with components
psi_m(x) :=
Therefore, if we have a quantum system defined in an arbitrary
Hilbert space in which a momentum operator is defined, the necessary
and sufficient condition for the existence of a spatial probability
interpretation of the system is the existence of a position operator
with commuting components which satisfy standard commutation
relations with the components of the momentum operator and the
angular momentum operator.
Thus we have reduced the existence of a probability interpretation
for particles in a bounded region of space to the question of the
existence of a position operator with the right properties.
We now investigate this existence problem for elementary particles,
i.e., objects represented by an irreducible representation of the
full Poincare group. We consider first the case of particles of
mass m>0, since the massless case needs additional considerations.
A. Massive case, m>0:
Let M := R^3 be the manifold of 3-momenta p. On the Hilbert space
H_m^d obtained by completion of the space of all C^infty functions
with compact support from M to the space C^d of d-component vectors
with complex entries, with inner product defined by
<phi|psi> := integral dp/sqrt(p^2+m^2) phi(p)^*psi(p),
we define the position operator
q := i hbar partial_p,
which satisfies the standard commutation relations, the momentum in
time direction,
p_0 := sqrt(m^2+|p|^2),
where m>0 is a fixed mass, and the operators
J := q x p + S,
K := (p_0 q + q p_0)/2 + p x S/(m+p_0),
where S is the spin vector in a unitaryirreducible representation of
so(3) on the vector space C^d of complex vectors of length d, with the
same commutation relations as J.
This is a unitary representation of the Poincare algebra; verification of the standard commutation relations (given, e.g., in Weinberg's Volume 1, p.61) is straightforward. It is not difficult to show that this representation is irreducible and extends to a representation of the full Poincare group. Obviously, this representation carries a position operator.
Since the physical irreducible representations of the Poincare group
are uniquely determined by mass and spin, we see that in the massive
case, a position operator must always exist. An explicit formula in
terms of the Poincare generators is obtained through division by m
in the formula
mq = K - ((K dot p) p/p_0 + J x p)/(m+p_0),
which is straightforward, though a bit tedious to verify from the above.
That (up to a constant shift) there is no other possibility follows from
Already the notion of a particle depends on the observer, as shown by
the Unruh effect. It is no surprise that the position of something
observer-dependent is also observer-dependent.
It explains naturally why position operators are necessarily
noninvariant under Lorentz boosts.
Quantum fields are covariant and exist everywhere, so they need neither observers nor a particular position operator. That this is not the case for particles is - in view of the fact that physical objects existed long before observers came into existence - sufficient reasons why particles cannot be fundamental.
Note also that in case of the Dirac equation, the position operator is _not_ the operator multiplying a solution psi(x) of the Dirac equation by the spacelike part of x (which would mix electron and positron states), but a related operator obtained by first applying a so-called Foldy-Wouthuysen transformation.
Let M_0 := R^3\{0} be the manifold of nonzero 3-momenta p, and let
p_0 := |p|, n := p/p_0.
The Hilbert space H_0^d (defined as before but now with m=0 and with
M_0 in place of M) obtained by completion of the space of all C^infty
functions with compact support from M to the space C^d of d-component
vectors with complex entries, with inner product defined by
<phi|psi> := integral d\p/sqrt(p^2) phi(p)^*psi(p),
carries a natural massless representation of the Poincare algebra,
defined by
J := q x p + S,
K := (p_0 q + q p_0)/2 + n x S,
where q = i hbar partial_p is the position operator, and S is the
spin vector in a unitary irreducible representation of so(3) on C^d,
with the same commutation relations as J.
Again, verification of the standard commutation relations is
straightforward. (Indeed, this representation is the limit of the
above massive representation for m --> 0.)
It is easily seen that the helicity
lambda := n dot S
is central in the (suitably completed) universal envelope of the
Lie algebra, and that the possible eigenvalues
of the helicity are s,s-1,...,-s, where s=(d-1)/2. Therefore, the
eigenspaces of the helicity operator carry by restriction unitary
representations of the Poincare algebra, which are easily seen to be
irreducible. They extend to a representation of the connected
Poincare group. Moreover, the invariant subspace H_s formed by the
direct sum of the eigenspaces for helicity s and -s form a massless
irreducible spin s representation of the full Poincare group.
(It is easy to see that changing K to K-t(p_0)p for an arbitrary differentiable function t of p_0 preserves all commutation relations, hence gives another representation of the Poincare algebra. Since the massless irreducible representations of the Poincare group are uniquely determined by their spin, the resulting representations are equivalent. This corresponds to the freedom below in choosing a position operator.)
Now suppose that a massless irreducible representation of fixed helicity has a position operator x satisfying the canonical commutation relations with p and the above commutator relations with J. By Wigner's classification, such a representation is unique up to isomorphism, and hence is isomorphic to the representation constructed above. (Working in this representation makes the arguments less technical.) Thus we may assume w.l.o.g. that some Poincare invariant subspace H of L^2(M_0)^d has a position operator x satisfying the canonical commutation relations with p and the above commutator relations with J.
On this subspace, F=q-x commutes with p, hence its components must be
a (possibly matrix-valued) function F(p) of p. Commutation with p
implies that partial_p x F = 0, and, since M_0 is simply connected,
that F is the gradient of a scalar function f. Rotation invariance
then implies that this function depends only on p_0=|p|. Thus
F = partial_p f(p_0) = f'(p_0) n.
Thus the position operator takes the form
x = q - f'(p_0) n.
In particular,
x x p = q x p.
(The letter x is here ambiguous, standing for the vector and the cross
product sign. I was too lazy to make all vectors fat....)
Now the algebra of linear operators on the dense subspace of C^infty
functions in H contains the components of p, J, K and x, hence those
of
J - x x p = J - q x p = S.
Thus the (p-independent) operators from the spin so(3) act on H.
But this implies that either H=0 (no helicity) or H = L^2(M_0)^d
(all helicities between s and -s).
Since the physical irreducible representations of the Poincare group
are uniquely determined by mass and spin, and for s>1/2, the spin s
Hilbert space H_s is a proper, nontrivial subspace of L^2(M_0)^d,
we proved the following theorem:
Theorem. An irreducible representations of the full Poincare group with mass m>=0 and finite spin has a position operator transforming like a 3-vector and satisfying the canonical commutation relations if and only if either m>0 or m=0 and s<=1/2 (but s=0 if only the connected poincare group is considered).
This theorem was announced without giving details in
As a consequence of our discussion, photons (m=0, s=1) and gravitons (m=0, s=2) cannot be given natural probabilities for being in any given bounded region of space. Chiral spin 1/2 particles also do not have a position operator and hence have no such probabilities, by the same argument, applied to the connected Poincare group.
(Note that measured are only frequencies, intensities and
S-matrix elements; these don't need a well-defined position concept
but only a well-defined momentum concept, from which frequencies
can be found via omega=p_0/hbar - since c=1 in the present setting,
and directions via n = p/p_0.)
However, assuming there are scalar massless Higgs particles (s=0),
one could combine such a Higgs, a photon, and a graviton into
a single reducible representation on L^2(M_0)^5, using the above
construction. By our derivation, one can find position eigenstates
which are superpositions of Higgs, photon, and graviton. Thus to
be able to regard photons and gravitons as particles with a proper
probability interpretation, one must consider Higgs, photons, and
gravitons as aspects of the same localizable particle, which we
might call a graphoton. (Without gravity, a phiggs particle would
also do.)
If the concept of an observable is not tied to that of a Hermitian operator but rather to that of a POVM (positive operator-valued measure), there is more flexibility, and covariant POVMs for positon measurements can be meaningfully defined, even for photons. See, e.g.,
Therefore, the concept of a photon position is necessarily subjective, since it depends on the POVM used, hence on the way the measurement is performed. It does not describe something objective.
The POVM does not allow one to talk about the position of a photon
- which could exist only if the corresponding operator existed -,
but only about the measured position: The photon is somewhere near the
range of values established by the measurement, without any more
definite statement being possible. On the other hand, for observables
corresponding to Hermitian operators, there are states in which
a definite statement is (at least theoretically) possible that the
observable has a value in a given range.
Papers related to position operators:
There are a few papers by M. Hawton, e.g., this and this on a nonstandard position operator which does not transform like a 3-vector. This is unphysical since it does not give orientation-independent probabilities for observing a photon in a spherical region of space. Claims to the contrary, supposedly constructing a Lorentz invariant photon number density, are erroneous.
Other nonstandard position operators violating the conditions necessary for a probability interpretation were discussed earlier, starting with
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ