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Quantum Field Operators

We may expect symmetry between particles and antiparticles. The interpretation of antiparticles as negative energy particles going backwards in time motivates the definition of the quantum field operator».


We will find stronger reasons, to do with satisfying locality, for thinking that interaction operators are composed of field operators with this general form. The Hermitian conjugate of a quantum field operator, has the reverse effect, creating a particle or annihilating an antiparticle,

While quantum field operators defined here are formally those of standard quantum field theory», the interpretation adheres more closely to the views of Dirac and Feynman than to the “modern view” that fields are the fundamental objects of Nature. I have not used canonical quantization» to convert from a classical theory or second quantization^» of a field.

Normal Order

The interaction of a particle will be modelled by the annihilation of the old state of the particle and the creation of a new state. This means that all annihilation operators must be placed to the right of all creation operators in the interaction density, I(x). An operator for which this is so is in normal order». Putting an operator into normal order is normal ordering, and is denoted by colons.


In particular, the normal ordered product of fields is
where a plus is used for Bosons and a minus for Fermions.

The Dirac Field Operator

Since we can measure the position of an electron, it must be possible to form a projection operator for position at given time,
from the Hamiltonian density and a suitable configuration of matter (more accurately, X_P(x) is summed over a small range of positions depending on the resolution of measurement). We may expect symmetry between particles and antiparticles. The interpretation of antiparticles as negative energy particles going backwards in time indicates the replacement of negative energy annihilation operators with creation operators. This motivates the definition of the Dirac field operator», which annihilates a particle or creates the corresponding antiparticle.


The observable quantity, current density, uses the Dirac adjoint, so we expect the adjoint operators to appear alongside the field operators in the Hamiltonian density. The Dirac adjoint of the annihilation operator creates a particle,
The Dirac adjoint of the antiparticle creation operator annihilates an antiparticle,
The Dirac adjoint of the field operator creates a particle or annihilates an antiparticle,

Locality of Dirac Field Operators

Since Dirac particles are Fermions we have anticommutation relations for the Dirac field operator.
Dirac field operators will appear in pairs in the Hamiltonian density in such a way as to ensure commutation relations required of the locality condition.


Proof.

The Current Density Observable

For electrons, current is an observable quantity. Since measurement is always the result of interactions between matter, a Hermitian operator, j, whose expectation is the classical electrical current must appear in the Hamiltonian density. To ensure that locality is satisfied, current is composed of Dirac field operators.


Then, given the particle state in H¹ with ,
in agreement with current density for a single particle state. For an antiparticle states, spin indices are transposed. Transposition is equivalent to pre- and post-multiplying γ by a matrix,
which has the effect on the wave function of reversing the order of the spin indices. Thus, a negative energy spin down electron will appear as a positive energy spin up positron.

The Dirac Field Operator ↑ The Photon Field Operator →