Fields giving rise to fermionic particles
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 or a
Dirac fermion
; observing
neutrinoless double-beta decay
experimentally would settle this question.
Basic properties
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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
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The prominent example of a spin-1/2 fermion field is the
Dirac field
(named after
Paul Dirac
), and denoted by
. The equation of motion for a free spin 1/2 particle is the
Dirac equation
,
where
are
gamma matrices
and
is the mass. The simplest possible solutions
to this equation are plane wave solutions,
and
. These
plane wave
solutions form a basis for the Fourier components of
, allowing for the general expansion of the wave function as follows,
u
and
v
are spinors, labelled by spin,
s
and spinor indices
. 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
,
is promoted to an operator, so the coefficients of its Fourier modes must be operators too. Hence,
and
are operators. The properties of these operators can be discerned from the properties of the field.
and
obey the anticommutation relations:
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
and
, the anticommutation relations for the coefficients can be computed.
In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that
creates a fermion of momentum
p
and spin s, and
creates an antifermion of momentum
q
and spin
r
. The general field
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,
, 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
. This makes the reason for the choice of
clear. This is because the general Lorentz transform on
is not
unitary
so the quantity
would not be invariant under such transforms, so the inclusion of
is to correct for this. The other possible non-zero
Lorentz invariant
quantity, up to an overall conjugation, constructible from the fermionic fields is
.
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.
Such an expression has its indices suppressed. When reintroduced the full expression is
The
Hamiltonian
(
energy
) density can also be constructed by first defining the momentum canonically conjugate to
, called
With that definition of
, the Hamiltonian density is:
where
is the standard
gradient
of the space-like coordinates, and
is a vector of the space-like
matrices. It is surprising that the Hamiltonian density doesn't depend on the time derivative of
, directly, but the expression is correct.
Given the expression for
we can construct the Feynman
propagator
for the fermion field:
we define the
time-ordered
product for fermions with a minus sign due to their anticommuting nature
Plugging our plane wave expansion for the fermion field into the above equation yields:
where we have employed the
Feynman slash
notation. This result makes sense since the factor
is just the inverse of the operator acting on
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
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
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References
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]
- 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
.
S2CID
120108219
.
- Peskin, M and Schroeder, D. (1995).
An Introduction to Quantum Field Theory
, Westview Press. (See pages 35?63.)
- Srednicki, Mark (2007).
Quantum Field Theory
Archived
2011-07-25 at the
Wayback Machine
, Cambridge University Press,
ISBN
978-0-521-86449-7
.
- Weinberg, Steven (1995).
The Quantum Theory of Fields
, (3 volumes) Cambridge University Press.