In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the Rarita–Schwinger equation is the
relativistic field equation
In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equ ...
of
spin-3/2
fermions. It is similar to the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
for spin-1/2 fermions. This equation was first introduced by
William Rarita and
Julian Schwinger
Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
in 1941.
In modern notation it can be written as:
:
where
is the
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
,
and
are
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
,
is the mass,
,
and
is a vector-valued
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
with additional components compared to the four component spinor in the Dirac equation. It corresponds to the
representation of the Lorentz group, or rather, its part.
This field equation can be derived as the
Euler–Lagrange equation corresponding to the Rarita–Schwinger
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:
:
where the bar above
denotes the
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 adjoi ...
.
This equation controls the propagation of the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
of composite objects such as the
delta baryon
The Delta baryons (or baryons, also called Delta resonances) are a family of subatomic particle made of three up or down quarks (u or d quarks), the same constituent quarks that make up the more familiar protons and neutrons.
Properties
Four c ...
s () or for the conjectural
gravitino
In supergravity theories combining general relativity and supersymmetry, the gravitino () is the gauge fermion supersymmetric partner of the hypothesized graviton. It has been suggested as a candidate for dark matter.
If it exists, it is a f ...
. So far, no
elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, a ...
with spin 3/2 has been found experimentally.
The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation
, where
is an arbitrary spinor field. This is simply the local
supersymmetry of
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
, and the field must be a gravitino.
"Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
Equations of motion in the massless case
Consider a massless Rarita–Schwinger field described by the Lagrangian density
:
where the sum over spin indices is implicit,
are Majorana spinors, and
:
To obtain the equations of motion we vary the Lagrangian with respect to the fields
, obtaining:
:
using the Majorana flip properties
we see that the second and first terms on the RHS are equal, concluding that
:
plus unimportant boundary terms.
Imposing
we thus see that the equation of motion for a massless Majorana Rarita–Schwinger spinor reads:
:
Drawbacks of the equation
The current description of massive, higher spin fields through either Rarita–Schwinger or
Fierz–Pauli formalisms is afflicted with several maladies.
Superluminal propagation
As in the case of the Dirac equation, electromagnetic interaction can be added by promoting the partial derivative to
gauge covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are ...
:
:
.
In 1969, Velo and Zwanziger showed that the Rarita–Schwinger Lagrangian coupled to
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
leads to equation with solutions representing wavefronts, some of which propagate faster than light. In other words,
the field then suffers from acausal, superluminal propagation; consequently, the
quantization in interaction with electromagnetism is essentially flawed. In extended supergravity, though, Das and Freedman
[; ] have shown that local supersymmetry solves this problem.
References
Sources
*
* Collins P.D.B.,
Martin A.D., Squires E.J., ''Particle physics and cosmology'' (1989) Wiley, ''Section 1.6''.
*
*
*
{{DEFAULTSORT:Rarita-Schwinger equation
Quantum field theory
Spinors
Partial differential equations
Fermions
Equations of physics
Mathematical physics