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In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields. The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor. It is not known whether the neutrino is a Majorana fermion
Majorana fermion
or a Dirac fermion; observing neutrinoless double-beta decay experimentally would settle this question.

Contents

1 Basic properties 2 Dirac fields 3 See also 4 References

Basic properties[edit] Free (non-interacting) fermionic fields obey canonical anticommutation relations; i.e., involve the anticommutators a, b = ab + ba, rather than the commutators [a, b] = ab − ba of bosonic or standard quantum mechanics. Those relations also hold for interacting fermionic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states. It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta. They also result in the Pauli exclusion principle: two fermionic particles cannot occupy the same state at the same time. Dirac fields[edit] The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul Dirac), and denoted by

ψ ( x )

displaystyle psi (x)

. The equation of motion for a free spin 1/2 particle is the Dirac equation,

(

i

γ

μ

μ

− m

)

ψ ( x ) = 0.

displaystyle left(igamma ^ mu partial _ mu -mright)psi (x)=0.,

where

γ

μ

displaystyle gamma ^ mu

are gamma matrices and

m

displaystyle m

is the mass. The simplest possible solutions to this equation are plane wave solutions,

ψ

1

( x ) = u ( p )

e

− i p . x

displaystyle psi _ 1 (x)=u(p)e^ -ip.x ,

and

ψ

2

( x ) = v ( p )

e

i p . x

displaystyle psi _ 2 (x)=v(p)e^ ip.x ,

. These plane wave solutions form a basis for the Fourier components of

ψ ( x )

displaystyle psi (x)

, allowing for the general expansion of the wave function as follows,

ψ ( x ) = ∫

d

3

p

( 2 π

)

3

1

2

E

p

s

(

a

p

s

u

s

( p )

e

− i p ⋅ x

+

b

p

s †

v

s

( p )

e

i p ⋅ x

)

.

displaystyle psi (x)=int frac d^ 3 p (2pi )^ 3 frac 1 sqrt 2E_ p sum _ s left(a_ mathbf p ^ s u^ s (p)e^ -ipcdot x +b_ mathbf p ^ sdagger v^ s (p)e^ ipcdot x right).,

u and v are spinors, labelled by spin, s. For the electron, a spin 1/2 particle, s = +1/2 or s=−1/2. The energy factor is the result of having a Lorentz invariant integration measure. In second quantization,

ψ ( x )

displaystyle psi (x)

is promoted to an operator, so the coefficients of its Fourier modes must be operators too. Hence,

a

p

s

displaystyle a_ mathbf p ^ s

and

b

p

s †

displaystyle b_ mathbf p ^ sdagger

are operators. The properties of these operators can be discerned from the properties of the field.

ψ ( x )

displaystyle psi (x)

and

ψ ( y

)

displaystyle psi (y)^ dagger

obey the anticommutation relations:

ψ

a

(

x

) ,

ψ

b

(

y

)

=

δ

( 3 )

(

x

y

)

δ

a b

,

displaystyle left psi _ a (mathbf x ),psi _ b ^ dagger (mathbf y )right =delta ^ (3) (mathbf x -mathbf y )delta _ ab ,

where a and b are spinor indices. We impose an anticommutator relation (as opposed to a commutation relation as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics. By putting in the expansions for

ψ ( x )

displaystyle psi (x)

and

ψ ( y )

displaystyle psi (y)

, the anticommutation relations for the coefficients can be computed.

a

p

r

,

a

q

s †

=

b

p

r

,

b

q

s †

= ( 2 π

)

3

δ

3

(

p

q

)

δ

r s

,

displaystyle left a_ mathbf p ^ r ,a_ mathbf q ^ sdagger right =left b_ mathbf p ^ r ,b_ mathbf q ^ sdagger right =(2pi )^ 3 delta ^ 3 (mathbf p -mathbf q )delta ^ rs ,,

In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that

a

p

s †

displaystyle a_ mathbf p ^ sdagger

creates a fermion of momentum p and spin s, and

b

q

r †

displaystyle b_ mathbf q ^ rdagger

creates an antifermion of momentum q and spin r. The general field

ψ ( x )

displaystyle psi (x)

is now seen to be a weighted (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field,

ψ ¯

 

=

d e f

 

ψ

γ

0

displaystyle overline psi stackrel mathrm def = psi ^ dagger gamma ^ 0

, is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions. With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity

ψ ¯

ψ

displaystyle overline psi psi ,

. This makes the reason for the choice of

ψ ¯

=

ψ

γ

0

displaystyle overline psi =psi ^ dagger gamma ^ 0

clear. This is because the general Lorentz transform on ψ is not unitary so the quantity

ψ

ψ

displaystyle psi ^ dagger psi

would not be invariant under such transforms, so the inclusion of

γ

0

displaystyle gamma ^ 0 ,

is to correct for this. The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructible from the fermionic fields is

ψ ¯

γ

μ

μ

ψ

displaystyle overline psi gamma ^ mu partial _ mu psi

. Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler–Lagrange equation of the system recover the Dirac equation.

L

D

=

ψ ¯

(

i

γ

μ

μ

− m

)

ψ

displaystyle mathcal L _ D = overline psi left(igamma ^ mu partial _ mu -mright)psi ,

Such an expression has its indices suppressed. When reintroduced the full expression is

L

D

=

ψ ¯

a

(

i

γ

a b

μ

μ

− m

I

a b

)

ψ

b

displaystyle mathcal L _ D = overline psi _ a left(igamma _ ab ^ mu partial _ mu -mmathbb I _ ab right)psi _ b ,

The Hamiltonian (energy) density can also be constructed by first defining the momentum canonically conjugate to

ψ ( x )

displaystyle psi (x)

, called

Π ( x ) :

displaystyle Pi (x):

Π  

=

d e f

 

L

D

∂ (

0

ψ )

= i

ψ

.

displaystyle Pi overset mathrm def = frac partial mathcal L _ D partial (partial _ 0 psi ) =ipsi ^ dagger ,.

With that definition of

Π

displaystyle Pi

, the Hamiltonian density is:

H

D

=

ψ ¯

[

− i

γ →

∇ →

+ m

]

ψ

,

displaystyle mathcal H _ D = overline psi left[-i vec gamma cdot vec nabla +mright]psi ,,

where

∇ →

displaystyle vec nabla

is the standard gradient of the space-like coordinates, and

γ →

displaystyle vec gamma

is a vector of the space-like

γ

displaystyle gamma

matrices. It is surprising that the Hamiltonian density doesn't depend on the time derivative of

ψ

displaystyle psi

, directly, but the expression is correct. Given the expression for

ψ ( x )

displaystyle psi (x)

we can construct the Feynman propagator for the fermion field:

D

F

( x − y ) =

0

T ( ψ ( x )

ψ ¯

( y ) )

0

displaystyle D_ F (x-y)=leftlangle 0leftT(psi (x) overline psi (y))right0rightrangle

we define the time-ordered product for fermions with a minus sign due to their anticommuting nature

T

[

ψ ( x )

ψ ¯

( y )

]

 

= def

  θ

(

x

0

y

0

)

ψ ( x )

ψ ¯

( y ) − θ

(

y

0

x

0

)

ψ ¯

( y ) ψ ( x ) .

displaystyle Tleft[psi (x) overline psi (y)right] overset text def = theta left(x^ 0 -y^ 0 right)psi (x) overline psi (y)-theta left(y^ 0 -x^ 0 right) overline psi (y)psi (x).

Plugging our plane wave expansion for the fermion field into the above equation yields:

D

F

( x − y ) = ∫

d

4

p

( 2 π

)

4

i (

p

/

+ m )

p

2

m

2

+ i ϵ

e

− i p ⋅ ( x − y )

displaystyle D_ F (x-y)=int frac d^ 4 p (2pi )^ 4 frac i( p!!!/ +m) p^ 2 -m^ 2 +iepsilon e^ -ipcdot (x-y)

where we have employed the Feynman slash notation. This result makes sense since the factor

i (

p

/

+ m )

p

2

m

2

displaystyle frac i( p!!!/ +m) p^ 2 -m^ 2

is just the inverse of the operator acting on

ψ ( x )

displaystyle psi (x)

in the Dirac equation. Note that the Feynman propagator for the Klein–Gordon field has this same property. Since all reasonable observables (such as energy, charge, particle number, etc.) are built out of an even number of fermion fields, the commutation relation vanishes between any two observables at spacetime points outside the light cone. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. We have therefore correctly implemented Lorentz invariance
Lorentz invariance
for the Dirac field, and preserved causality. More complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics) can be analyzed too, by various perturbative and non-perturbative methods. Dirac fields are an important ingredient of the Standard Model. See also[edit]

Dirac equation Einstein–Maxwell–Dirac equations Spin-statistics theorem Spinor

References[edit]

Edwards, D. (1981). "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories". Int. J. Theor. Phys. 20 (7): 503–517. Bibcode:1981IJTP...20..503E. doi:10.1007/BF00669437.  Peskin, M and Schroeder, D. (1995). An Introduction to Quantum Field Theory, Westview Press. (See pages 35–63.) Srednicki, Mark (2007). Quantum Field Theory, Cambridge University Press, ISBN 978-0-521-86449-7. Weinberg, Steven (1995). The Quantum Theory of Fields, (3 volumes) Cambridge Unive

.