Probability Density in Relativistic Quantum Mechanics
and Quantum Field Theory
Subtitled: Whence the Relativistic Wave Solution Normalization Factors?
1 Relativistic Free Scalar Field Equation
For a free scalar field governed by the (relativistic) Klein-Gordon equation
|
|
and its complex conjugate (transpose)
|
|
(2) |
we have solutions
|
|
and
|
|
The summation is from infinite k in the negative x direction to infinite k in the positive x direction plus similar summations for the y and
z directions. The reasons for the normalization
factors in the solutions will be explained herein
later on.
Solutions (3) and (4) arise for problems with boundary conditions, as such problems entail discrete values for wave number (equivalently, wavelength or 3-momentum k.) Each term in the summation has a discrete value of 3-momentum k.
There are also continuous solutions, comparable to (3) and (4), but containing integrals over 3-momentum, rather than summations. These are Fourier integrals (representing wave packets) rather than Fourier series. They are generally used for cases without boundary conditions.
Note that only one term in the discrete summation solution can be used in problems where the particle in question can be approximated by a single value for 3 momentum (and thus a pure complex sinusoid extending to infinity.)
While we shall only address the discrete solutions here, the reader should keep in mind that a parallel development exists for continuous solutions.
2 Relativistic Quantum Mechanics vs Quantum Field Theory
2.1 Difference Between Solutions in RQM and QFT
In relativistic quantum mechanics (RQM), (x) of (3)
[with b†(k) = 0 for all k] represents a single
particle general state that is a sum of discrete momentum eigenstates of that
single particle. The coefficients a(k)
are numbers, amplitudes which as we will see, when squared, equal the probability
of finding the single particle in that discrete eigenstate.
In quantum field theory (QFT), on the other hand, the coefficients a(k) and a†(k) in (3) and (4) are not numbers but operators that each destroy or create single particle eigenstates.
Until we note otherwise near the end, when we wish to extrapolate our results to QFT, we shall restrict the development herein to RQM (and thus make it easier to understand).
2.2 Definition of Eigensolutions
As noted, the solution (3) in RQM is for single particle general (sum of eigenstates) state, and does not contain operators. Thus, from henceforth while discussing RQM, we substitute the numerical symbol Ak for a(k), as the latter symbol is normally used in QFT to represent an operator. Hence, in RQM, the K-G solutions are of form
|
|
where k has unit norm.
That is, we define the eigenstate with symbol
k as
|
|
so that
|
|
(7) |
or more generally,
|
|
3 Probability Density
3.1 Review of Non-Relativistic QM Probability Density
In non-relativistic quantum mechanics (NRQM), we encountered 1) the wave function solution to the Schroedinger equation Ψ, and 2) the particle probability density ρ = Ψ*Ψ (or equivalently for a scalar quantity, Ψ†Ψ) We review here the derivation of that relation for probability density.
The general procedure:
Use the governing wave equation to deduce another equation having the form of the continuity equation
|
|
and we will then know that ρ, whatever it turns out to be, must represent a conserved quantity. Its integral over all space is constant in time. If we normalize ρ such that when integrated over all space, the result equals one, we can conjecture that ρ is the particle probability density (which when integrated over all space equals the probability that we will find the particle somewhere in all space, i.e., one.) Then throughout time, as our particle evolves, moves, and rearranges its probability density distribution, the total probability of finding it somewhere in space is always one. It turns out, from experiment, that our conjecture that this quantity ρ in NRQM equals probability density is true.
Using the Schroedinger Equation:
First, pre-multiply the Shroedinger equation by the complex conjugate of the wave function
|
|
Then, post-multiply the complex conjugate of the Schroedinger equation by the wave function
|
|
where V is real so V=V†. Adding (10) to (11), we get
|
|
(12) |
or
|
|
This is the same as the continuity equation (9) if we take
|
|
as our probability density and
|
|
(15) |
as our probability current. This is how the commonly used relation (14) is found.
3.2 Probabilty Density in RQM
Using the Klein-Gordon Equation
For RQM, we start with the Klein-Gordon field equation (1)
rather than Schroedinger equation.
First pre-multiply it by †,
i.e.,
|
|
Then take the complex conjugate Klein-Gordon equation
post multiplied by ,
i.e.,
|
|
and subtract the former (16) from the latter (17). The result (after multiplying by i) and using the identity shown under (13) is
|
|
(18) |
This has the form of the continuity equation
|
|
where we introduce 4D vector notation. The probability density for a Klein-Gordon particle is then
|
|
with the probability current
|
|
(21) |
Importantly, and perhaps surprisingly, the relativistic form of the probability density (20) is not the same as (14), the non-relativistic probability density.
Probability for Discrete Solutions
For a single particle state in RQM, the probability density (20) in terms of the solution (5) is
|
|
When we integrate over the volume V, all terms with k/ ≠ k go to zero, leaving
|
|
and thus |Ak|2 is the probability of measuring the kth eigenstate.
Reason for Normalization Factors
Note that obtaining (23) is
the reason for the normalization factors 1/ used in the solution
. Those factors result in a total probability
of one for a single particle and |Ak|2 as the
probability for measuring the kth
state. That is, the form of the
relativistic field equation gave us the form of the probability density in (20)
and (22). The time derivatives in (20)
gave us a factor of ωk,
and the two terms a factor of 2. These
cancel in (23)
with the 2ωk
in the denominators. The V term in the denominator cancels in the
integration over volume in (23) and the result is a
total probability of 1.
Difference from NRQM
Note that
|
|
because in RQM (unlike in NRQM), the LHS of (24) does not represent the integral of the probability density over space.
Relativistic Invariance of Probability
This probability value of unity in (23) is a relativistic invariant. If we change our frame, the energy spectrum (i.e., the ωk values) will change (K.E. looks different for each energy-momentum eigenstate). But these changes cancel out in the probability calculation and always result in a probability of one for any frame. Further, the Ak here are constants that do not vary with frame, so the probability of finding any particular state is also independent of what frame the measurements are taken in.
3.3
Spin 
For spin (Dirac spinor) and spin 1 (vector) particles,
one proceeds in similar fashion to deduce the continuity equation from the
corresponding wave equation.
Dirac particles
Pre-multiply the Dirac equation by
|
|
(25) |
and add to the complex conjugate transpose Dirac equation post-multiplied by ψ
|
|
(26) |
to yield
|
|
Thus, the probability density for a relativistic spin particle is
|
|
(28) |
with the probability current readily deduced from (27). The four current is then
|
|
(29) |
Vector Particles
For spin 1 massless (massive) particles, one uses the Maxwell (Proça) wave equation and proceeds in a manner similar to that used above for Klein-Gordon particles, except that now our pre and post multiplications must be inner products with the photon wave solution Aα. It is a good exercise to work this out to find the result
|
|
with 4 current
|
|
(31) |
All this works if we consider the photon solution having the form of (5).
Caveat (Subtler, Advanced Refinement of Understanding)
One finds later, however, that any particle which is its
own antiparticle, such as the photon, has the property (or for photons,
), as the action of taking the complex
conjugate effectively switches charge (if you haven’t studied this yet, don’t
worry about how right now). For this to
be true, the wave equation solution must have the form
|
|
where C is a real number, rather than the more general form of (3).
Since the photon is its own anti-particle, that means (30) equals zero. This led early researchers to conclude that probability density as calculated herein is actually charge density (or at least proportional to charge density), as the photon has zero charge.
However, as long as we confine ourselves to RQM, where, like NRQM, we can consider there to be only particles and no antiparticles, we can take the photon to have the form of (5), i.e., (32) without the second term on the RHS. For this case, (30) can be considered the photon probability density.
Subtleties such as this, as well as a number of others, are best handled with QFT, rather than RQM.
4 Probability Density in Quantum Field Theory
QFT differs from RQM in that the coefficients a(k),
b(k), and their complex conjugates in solutions (3) and
(4)
turn out to be operators that create and destroy particles and antiparticles,
rather than numeric quantities. The
particles and antiparticles are then represented as kets, such as |
,
while the solutions
and
†
are operator fields (often simply
referred to as fields.)
QFT also is far more amenable to multiparticle states
than NRQM or RQM, and such states are represented with ket forms such as |3k, Ak’α
(denoting 3 scalars all with momentum k and one photon with momentum k/.)
QFT also does not typically deal with probability densities, as that is something more suitable for a single particle state. For two electrons, for example, the total probability of finding an electron somewhere in space would be two. For this and other reasons, QFT deals primarily with number of particles, rather than probability density for a given particle.
However, for purposes of edification, clarification, and ready comparison of QFT to RQM and NRQM, we will herein illustrate the QFT equivalent of probability density.
Probability Density Operator in QFT
One starts with the Klein-Gordon equation, its complex conjugate and the solutions (3) and (4). The same steps (16) to (20) are then followed, except that 1) we are not using the solution form (5) because we retain the terms with b(k) and b†(k), and 2) the final result will be a density operator, not a simple numeric density. We will then need to find the expectation value of that operator for a particular particle (ket) in order to get a numerical probability density.
Proceeding as before in steps (16) to (20), we end up with a relation similar to (20)
|
|