The Quantum Lagrangian:

The Dirac Form

In the classical case the Lagrangian function comes from the dot product of a particle’s dynamic properties (the momentum-energy four-vector) and the particle’s kinematic properties (the spacetime distance-duration four-vector). The number that we calculate is a Lorentz invariant: it has the same value regardless of the inertial frame in which the measurements that go into the calculation are made. Thus, our Lagrangian formulation of dynamics is valid in all inertial frames.

We can’t use that product in devising a quantum version of the Lagrangian, because those particular variables, when taken together, run afoul of Heisenberg’s indeterminacy. Multiplying the momentum-energy four-vector by itself gets around that limitation and still gives us a Lorentz invariant,

(Eq’n 1)

But integrating that equation with respect to time does not give us an action that we can put through an Euler-Lagrange equation to get an equation of motion or, in the case of a field, an equation of evolution. To convert that Lorentz invariant into something that will give us an action, we follow Dirac and rewrite it as

(Eq’n 2)

That looks like something that we could factor into an expression that looks like

(Eq’n 3)

with

(Eq’n 4)

Noting that the order of multiplication makes a difference in this case, we have Equation 3 as

(Eq’n 5)

so we multiply E by a coefficient a_{4} such that a_{4}^{2}=1.
We can represent the last term in that equation with a matrix,

(Eq’n 6)

For that product matrix to yield Equation 2 we must have as true to the calculation

(Eq’n 7)

in which the delta represents Kronecker’s delta expressed as a matrix; specifically, the identity matrix (I).

In order to satisfy the requirement of Equation 7, the a-coefficients must be matrices and they must be at least 4x4 matrices. They are, in fact, the gamma matrices of relativistic quantum theory. We thus have Equation 3, divided by minus one for convenience, as

(Eq’n 8)

That equation is satisfied if either of the factors zeroes out. As Dirac discovered, one factor gives us an equation describing matter and the other describes antimatter. But those factors are only the operators. They must have a state function on which to operate in order for us to make a calculation of expectation values. Because of the matrices, we must use a state function that has four components in the manner of a vector, which means that the state function is a spinor that looks like this,

(Eq’n 9)

In that array the top two entries encode the spin eigenstates (spin up and
spin down) of a spin-1/2 particle and the lower two entries encode the spin
eigenstates of the particles’s antimatter counterpart. So for the Dirac equation
describing matter, we write (using p_{μ}=-iℏ∂_{μ})

(Eq’n 10)

We have to use matrices in these equations because of the four-dimensional nature of spacetime and the fact that many of our calculations involve dot products of four-vectors. That nature also necessitates that particles possess a property called spin, in order to uphold the law of conservation of angular momentum. The matrices themselves represent the spin and the spinor represents the orientation state (spin up, spin down) in which the particle is likely to be found.

The operator in Equation 10 has units of energy. If we apply Born’s theorem,

(Eq’n 11)

we obtain some constant, a Lorentz invariant associated with the particle. If we integrate the constant over some temporal interval, we get a number with units of action. In the integrand we can think of the operator acting on the state function as writing a number and the Hermitian conjugate state function (ø*, the complex conjugate of the transpose) as reading it. The writing may produce features that the reading leaves out, so if we vary the reading field, the number thus produced will change. If the conjugate state function reads out the Dirac energy alone, then varying it will bring other features of the aleatric field into the calculation and make the number bigger. Thus, we state that the aleatric field must have such a form that the calculation of its Dirac energy must be a minimum with respect to variations in the conjugate field; that is,

(Eq’n 12)

When we work through the calculus of variations on that equation, we see that
the associated Euler-Lagrange equation is just Equation 10. The Lagrangian
(actually Lagrangian __density__) in that statement has the form

(Eq’n 13)

That function describes particles of matter, which carry half-integer Dirac units (usually one-half aitch-bar) of spin.

Equations 10 and 13 also apply to spin-3/4 particles. In that case we know the former as the Rarita-Schwinger equation and it uses spin matrices different from those used in the Dirac equation. Further, the associated spinor has eight elements instead of four, because of the more numerous spin eigenstates available to the particle and its antimatter counterpart.

The quantum theory also describes forcefields, which involve particles of their own, and their interactions with matter. Accounting for those will take us to the Proca form.

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