physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, specifically field theory and

particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...

, the Proca action describes a

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...

ive

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

-1

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of mass ''m'' in

Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...

. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca. The Proca equation is involved in the

Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...

and describes there the three massive

vector boson In particle physics, a vector boson is a boson whose spin equals one. The vector bosons that are regarded as elementary particles in the Standard Model are the gauge bosons, the force carriers of fundamental interactions: the photon of electromag ...

s, i.e. the Z and W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of

4-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...

Lagrangian density

The field involved is a complex

4-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. Whe ...

B^\mu = \left (\frac, \mathbf \right)

, where

\phi

is a kind of generalized

electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...

and

\mathbf

is a generalized

magnetic potential Magnetic potential may refer to: * Magnetic vector potential, the vector whose curl is equal to the magnetic B field * Magnetic scalar potential Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...

. The field

B^\mu

transforms like a complex four-vector. The Lagrangian density is given by: :

\mathcal=-\frac(\partial_\mu B_\nu^*-\partial_\nu B_\mu^*)(\partial^\mu B^\nu-\partial^\nu B^\mu)+\fracB_\nu^* B^\nu.

where

c

is the

speed of light in vacuum The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...

\hbar

is the reduced Planck constant, and

\partial_

is the

4-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and re ...

Equation

The Euler–Lagrange equation of motion for this case, also called the Proca equation, is: :

\partial_\mu(\partial^\mu B^\nu - \partial^\nu B^\mu)+\left(\frac\right)^2 B^\nu=0

which is equivalent to the conjunction ofMcGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, :

^\nu=0

with (in the massive case) :

\partial_\mu B^\mu=0 \!

which may be called a generalized

Lorenz gauge condition In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...

. For non-zero sources, with all fundamental constants included, the field equation is: :

c=\left( \left( +/ \right)- \right)

When

m = 0

, the source free equations reduce to

Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...

without charge or current, and the above reduces to Maxwell's charge equation. This Proca field equation is closely related to the Klein–Gordon equation, because it is second order in space and time. In the

vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

notation, the source free equations are: :

\Box \phi - \frac \left(\frac\frac + \nabla\cdot\mathbf\right) =-\left(\frac\right)^2\phi \!

\Box \mathbf + \nabla \left(\frac\frac + \nabla\cdot\mathbf\right) =-\left(\frac\right)^2\mathbf\!

and

\Box

is the D'Alembert operator.

Gauge fixing

The Proca action is the

gauge-fixed In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...

version of the

Stueckelberg action In field theory, the Stueckelberg action (named after Ernst Stueckelberg) describes a massive spin-1 field as an R (the real numbers are the Lie algebra of U(1)) Yang–Mills theory coupled to a real scalar field φ. This scalar field takes on ...

via the

Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...

. Quantizing the Proca action requires the use of

second class constraints A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishi ...

. If

m \neq 0

, they are not invariant under the gauge transformations of electromagnetism :

B^\mu \rightarrow B^\mu - \partial^\mu f

where

f

is an arbitrary function.

Lagrangian density

Equation

Gauge fixing

See also

References

Further reading