Conservation of Probability

In the quantum theory we express the equations of motion in terms of probabilities derived from the state function encoding a full description of the system. Those probabilities spread throughout space as probability densities (via Born’s theorem) and evolve with the elapse of time. We know that at any given instant the integral of those probability densities over the entire volume of space must equal the number one, indicating that the system certainly exists somewhere in space in some state of existence. The question that we must answer is this: do our fundamental quantum equations lead necessarily to the satisfaction of that requirement? Do those equations yield descriptions of quantum systems that conform to the conservation of probability?

If probability can spread in the likeness of a forcefield, then the analogue of a conservation law must take the same form as does the conservation law for other entities. We must therefore have an equation of continuity,

(Eq’n 1)

in which rho represents the probability density and the vector jay represents the probability current, the rate at which probability density flows through some unit of area, a probability flux.

To test that proposition in the non-relativistic approximation, we use Schrödinger’s equation (and its complex conjugate) in the absence of potentials (for simplicity),

(Eq’ns 2)

Born’s theorem tells us that

(Eq’n 3)

so Equation 1 becomes, upon substitution,

(Eq’n 4)

Replace the time derivatives in that equation by substituting from Equations 2 and get

(Eq’n 5)

But psi is a scalar function, so we know that

(Eq’n 6)

and we have Equation 5 as

(Eq’n 7)

which gives us the probability current as

(Eq’n 8)

The standard general solution of Schrödinger’s equation is a simple wave function,

(Eq’n 9)

If we substitute that expression into Equation 8, we get

(Eq’n 10)

If rho were to represent the density of electric charge, that last expression would describe the flux of electric current and Equation 1 would be the continuity equation describing conservation of electric charge. In this case that last expression describes the flux of probability current, indicating that Equation 4 is a proper continuity equation describing conservation of probability.

But that analysis only covers non-relativistic quantum mechanics. Is probability still conserved when our state function conforms to a relativistic quantum equation?

To answer that question, let’s look at the Klein-Gordon equation, which we obtain from the fundamental relativistic mass-momentum relationship,

(Eq’n 11)

That’s a wave equation, which has the general solution

(Eq’n 12)

which we can use with boundary conditions to obtain a specific solution for a given situation.

Subtract the mass term from both sides of Equation 11 and multiply the result from the left by ψ*, the complex conjugate of the state function. We get

(Eq’n 13)

From that equation subtract its complex conjugate and get

(Eq’n 14)

Substituting from Equation 12 into the time derivatives and dividing out the square of Dirac’s constant transforms that equation into

(Eq’n 15)

Dividing that equation by the numerical coefficient on the time derivative gives us

(Eq’n 16)

That’s the continuity equation with the probability flux conforming to Equation 8. Thus the Klein-Gordon equation yields a state function that conserves probability.

Now look at the Dirac equation, another example of a relativistic quantum equation,

(Eq’n 17)

In this case the state function is a four-component spinor, so the quantum momentum operator includes a set of 4x4 matrices (the gammas and the identity matrix I). Now write the Hermitian conjugate of that equation (the transpose of the complex conjugate),

(Eq’n 18)

In that equation we transpose the gamma matrix with the derivative of the spinor, which we represent in the manner of a row vector, just as we represent the unconjugated spinor in the manner of a column vector. I haven’t put a superscript star-tee on the gamma matrices because each gamma matrix is its own Hermitian conjugate.

Multiply Equation 17 on the left by the Hermitian conjugate of the state function, multiply Equation 18 on the right by the state function, and subtract the latter from the former. We get

(Eq’n 19)

If we separate the temporal and spatial components of that equation explicitly, we obtain

(Eq’n 20)

which looks like a continuity equation.

Equation 17 is relativistic, so we look at it through a relativistic perspective. The second term on the left side is a Lorentz invariant, so the first term must also be a Lorentz invariant (otherwise, the equation would not zero out properly). The momentum-energy operator gives us a proper four-vector, so the gamma matrices must also constitute the components of a four-vector (the first term in the equation being the equivalent of a four-vector dot product). That fact means that the gamma matrices must conform to a Lorentz Transformation.

In the Tenth Lecture of "Quantum Electrodynamics" Richard Feynman displayed the Lorentz Transformation of the gamma matrices as

(Eq’ns 21)

Of course, that’s only valid when we apply the gamma operators to a suitable
state function with its spinor. With that caveat in mind, we can find an
inertial frame in which γ_{x}’γ_{t}=0,
from which we get γ_{x}γ_{t}=v/c
or, more properly, when we multiply Equation 20 on the right by cγ_{4}/iℏ,

(Eq’n 22)

with

(Eq’n 23)

So Equation 20 is, indeed, a continuity equation for probability density.

The Dirac equation applies to particles carrying one half Dirac unit of spin. But the above analysis also applies to particles carrying three halves of a Dirac unit of spin. In that case we use Equation 17 with 8x8 gamma matrices and a state function with an 8-element spinor. That’s just a form of the Rarita-Schwinger equation. But those are particles of matter. What can we say of the force-bearing particles, which carry one full Dirac unit of spin? Can we get a continuity equation for probability density out of the Proca equation? Yes, we can, because the Proca equation can be put into a form identical to that of the Dirac equation: it merely uses a different set of gamma matrices and those don’t change the above analysis.

Thus we see that all of our fundamental quantum equations necessitate that the associated state function have such a form that the probability calculated from it will obey a conservation law.

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