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 eleme ...

ive spin-1 field of mass ''m'' in

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

. The corresponding equation is a

relativistic wave equation In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...

called the Proca equation. The Proca action and equation are named after Romanian physicist

Alexandru Proca Alexandru Proca (16 October 1897 – 13 December 1955) was a Romanian physicist who studied and worked in France. He developed the vector meson theory of nuclear forces and the Relativistic wave equations, relativistic quantum field equations tha ...

. The Proca equation is involved in the Standard Model and describes there the three massive vector bosons, i.e. the Z and W bosons. This article uses the (+−−−)

metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...

and

tensor index notation In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be c ...

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

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

, where

\phi

is a kind of generalized electric potential and

\mathbf

is a generalized magnetic potential. The field

B^\mu

transforms like a complex

four-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 ...

. The

Lagrangian density 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 ...

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,

\hbar

is the

reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...

, and

\partial_

is the 4-gradient.

Equation

The

Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

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. 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 The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant ...

, 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 subjec ...

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 In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...

Gauge fixing

The Proca action is the gauge-fixed version of the Stueckelberg action via the Higgs mechanism. Quantizing the Proca action requires the use of second class constraints. 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