Dirac adjoint

In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the Hermitian adjoint.

Possibly to avoid confusion with the usual Hermitian adjoint, some textbooks do not provide a name for the Dirac adjoint but simply call it "ψ-bar".

Definition

Let be a Dirac spinor. Then its Dirac adjoint is defined as

where denotes the Hermitian adjoint of the spinor , and is the time-like gamma matrix.

Spinors under Lorentz transformations

The Lorentz group of special relativity is not compact, therefore spinor representations of Lorentz transformations are generally not unitary. That is, if is a projective representation of some Lorentz transformation,

,

then, in general,

.

The Hermitian adjoint of a spinor transforms according to

.

Therefore, is not a Lorentz scalar and is not even Hermitian.

Dirac adjoints, in contrast, transform according to

.

Using the identity , the transformation reduces to

,

Thus, transforms as a Lorentz scalar and as a four-vector.

Usage

Using the Dirac adjoint, the probability four-current J for a spin-1/2 particle field can be written as

where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j:

.

Taking μ = 0 and using the relation for gamma matrices

,

the probability density becomes

.

See also

References

  • B. Bransden and C. Joachain (2000). Quantum Mechanics, 2e, Pearson. ISBN 0-582-35691-1.
  • M. Peskin and D. Schroeder (1995). An Introduction to Quantum Field Theory, Westview Press. ISBN 0-201-50397-2.
  • A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. ISBN 0-691-01019-6.
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