• Source: 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



ψ


{\displaystyle \psi }

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







ψ
¯



≡

ψ

†



γ

0




{\displaystyle {\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}}


where




ψ

†




{\displaystyle \psi ^{\dagger }}

denotes the Hermitian adjoint of the spinor



ψ


{\displaystyle \psi }

, and




γ

0




{\displaystyle \gamma ^{0}}

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



λ


{\displaystyle \lambda }

is a projective representation of some Lorentz transformation,




ψ
↦
λ
ψ


{\displaystyle \psi \mapsto \lambda \psi }

,
then, in general,





λ

†


≠

λ

−
1




{\displaystyle \lambda ^{\dagger }\neq \lambda ^{-1}}

.
The Hermitian adjoint of a spinor transforms according to





ψ

†


↦

ψ

†



λ

†




{\displaystyle \psi ^{\dagger }\mapsto \psi ^{\dagger }\lambda ^{\dagger }}

.
Therefore,




ψ

†


ψ


{\displaystyle \psi ^{\dagger }\psi }

is not a Lorentz scalar and




ψ

†



γ

μ


ψ


{\displaystyle \psi ^{\dagger }\gamma ^{\mu }\psi }

is not even Hermitian.
Dirac adjoints, in contrast, transform according to







ψ
¯



↦


(

λ
ψ

)


†



γ

0




{\displaystyle {\bar {\psi }}\mapsto \left(\lambda \psi \right)^{\dagger }\gamma ^{0}}

.
Using the identity




γ

0



λ

†



γ

0


=

λ

−
1




{\displaystyle \gamma ^{0}\lambda ^{\dagger }\gamma ^{0}=\lambda ^{-1}}

, the transformation reduces to







ψ
¯



↦



ψ
¯




λ

−
1




{\displaystyle {\bar {\psi }}\mapsto {\bar {\psi }}\lambda ^{-1}}

,
Thus,






ψ
¯



ψ


{\displaystyle {\bar {\psi }}\psi }

transforms as a Lorentz scalar and






ψ
¯




γ

μ


ψ


{\displaystyle {\bar {\psi }}\gamma ^{\mu }\psi }

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





J

μ


=
c



ψ
¯




γ

μ


ψ


{\displaystyle J^{\mu }=c{\bar {\psi }}\gamma ^{\mu }\psi }


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





J

=
(
c
ρ
,

j

)


{\displaystyle {\boldsymbol {J}}=(c\rho ,{\boldsymbol {j}})}

.
Taking μ = 0 and using the relation for gamma matrices






(

γ

0


)


2


=
I


{\displaystyle \left(\gamma ^{0}\right)^{2}=I}

,
the probability density becomes




ρ
=

ψ

†


ψ


{\displaystyle \rho =\psi ^{\dagger }\psi }

.




See also


Dirac equation
Rarita–Schwinger equation




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.

Kata Kunci Pencarian:

• Paul A.M. Dirac
• Persamaan Schrödinger
• Ruang Hilbert
• Dirac adjoint
• Dirac equation
• Dirac delta function
• Dirac–von Neumann axioms
• Adjoint functors
• List of mathematical topics in quantum theory
• Rarita–Schwinger equation
• Dirac operator
• Lagrangian (field theory)
• Dirac algebra