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Matter content and Lagrangian

Matter content

The model consists of a complex scalar field ϕ ( x ) {\displaystyle \phi (x)} minimally coupled to a gauge field A μ ( x ) {\displaystyle A_{\mu }(x)} .

This article discusses the theory on flat spacetime R 1 , 3 {\displaystyle \mathbb {R} ^{1,3}} (Minkowski space) so these fields can be treated (naïvely) as functions ϕ : R 1 , 3 → C {\displaystyle \phi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} } , and A μ : R 1 , 3 → ( R 1 , 3 ) ∗ {\displaystyle A_{\mu }:\mathbb {R} ^{1,3}\rightarrow (\mathbb {R} ^{1,3})^{*}} . The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one. The gauge field is also known as a principal connection, specifically a principal U ( 1 ) {\displaystyle {\text{U}}(1)} connection.

Lagrangian

The dynamics is given by the Lagrangian density

L = ( D μ ϕ ) ∗ D μ ϕ − V ( ϕ ∗ ϕ ) − 1 4 F μ ν F μ ν = ( ∂ μ ϕ ) ∗ ( ∂ μ ϕ ) − i e ( ( ∂ μ ϕ ) ∗ ϕ − ϕ ∗ ( ∂ μ ϕ ) ) A μ + e 2 A μ A μ ϕ ∗ ϕ − V ( ϕ ∗ ϕ ) − 1 4 F μ ν F μ ν {\displaystyle {\begin{array}{lcl}{\mathcal {L}}&=&(D_{\mu }\phi )^{*}D^{\mu }\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\\&=&(\partial _{\mu }\phi )^{*}(\partial ^{\mu }\phi )-ie((\partial _{\mu }\phi )^{*}\phi -\phi ^{*}(\partial _{\mu }\phi ))A^{\mu }+e^{2}A_{\mu }A^{\mu }\phi ^{*}\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\end{array}}}

where

• F μ ν = ( ∂ μ A ν − ∂ ν A μ ) {\displaystyle F_{\mu \nu }=(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })} is the electromagnetic field strength, or curvature of the connection.
• D μ ϕ = ( ∂ μ ϕ − i e A μ ϕ ) {\displaystyle D_{\mu }\phi =(\partial _{\mu }\phi -ieA_{\mu }\phi )} is the covariant derivative of the field ϕ {\displaystyle \phi }
• e = − | e | < 0 {\displaystyle e=-|e|<0} is the electric charge
• V ( ϕ ∗ ϕ ) {\displaystyle V(\phi ^{*}\phi )} is the potential for the complex scalar field.

Gauge-invariance

This model is invariant under gauge transformations parameterized by λ ( x ) {\displaystyle \lambda (x)} . This is a real-valued function λ : R 1 , 3 → R . {\displaystyle \lambda :\mathbb {R} ^{1,3}\rightarrow \mathbb {R} .}

ϕ ′ ( x ) = e i e λ ( x ) ϕ ( x ) and A μ ′ ( x ) = A μ ( x ) + ∂ μ λ ( x ) . {\displaystyle \phi '(x)=e^{ie\lambda (x)}\phi (x)\quad {\textrm {and}}\quad A_{\mu }'(x)=A_{\mu }(x)+\partial _{\mu }\lambda (x).}

Differential-geometric view

From the geometric viewpoint, λ {\displaystyle \lambda } is an infinitesimal change of trivialization, which generates the finite change of trivialization e i e λ : R 1 , 3 → U ( 1 ) . {\displaystyle e^{ie\lambda }:\mathbb {R} ^{1,3}\rightarrow {\text{U}}(1).} In physics, it is customary to work under an implicit choice of trivialization, hence a gauge transformation really can be viewed as a change of trivialization.

Higgs mechanism

If the potential is such that its minimum occurs at non-zero value of | ϕ | {\displaystyle |\phi |} , this model exhibits the Higgs mechanism. This can be seen by studying fluctuations about the lowest energy configuration: one sees that the gauge field behaves as a massive field with its mass proportional to e {\displaystyle e} times the minimum value of | ϕ | {\displaystyle |\phi |} . As shown in 1973 by Nielsen and Olesen, this model, in 2 + 1 {\displaystyle 2+1} dimensions, admits time-independent finite energy configurations corresponding to vortices carrying magnetic flux. The magnetic flux carried by these vortices are quantized (in units of 2 π e {\displaystyle {\tfrac {2\pi }{e}}} ) and appears as a topological charge associated with the topological current

J t o p μ = ϵ μ ν ρ F ν ρ   . {\displaystyle J_{top}^{\mu }=\epsilon ^{\mu \nu \rho }F_{\nu \rho }\ .}

These vortices are similar to the vortices appearing in type-II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions.

Example

A simple choice of potential for demonstrating the Higgs mechanism is

V ( | ϕ | 2 ) = λ ( | ϕ | 2 − Φ 2 ) 2 . {\displaystyle V(|\phi |^{2})=\lambda (|\phi |^{2}-\Phi ^{2})^{2}.}

The potential is minimized at | ϕ | = Φ {\displaystyle |\phi |=\Phi } , which is chosen to be greater than zero. This produces a circle of minima, with values Φ e i θ {\displaystyle \Phi e^{i\theta }} , for θ {\displaystyle \theta } a real number.

Scalar chromodynamics

See also: scalar chromodynamics

This theory can be generalized from a theory with U ( 1 ) {\displaystyle U(1)} gauge symmetry containing a scalar field ϕ {\displaystyle \phi } valued in C {\displaystyle \mathbb {C} } coupled to a gauge field A μ {\displaystyle A_{\mu }} to a theory with gauge symmetry under the gauge group G {\displaystyle G} , a Lie group.

The scalar field ϕ {\displaystyle \phi } is valued in a representation space of the gauge group G {\displaystyle G} , making it a vector; the label of "scalar" field refers only to the transformation of ϕ {\displaystyle \phi } under the action of the Lorentz group, so it is still referred to as a scalar field, in the sense of a Lorentz scalar. The gauge-field is a g {\displaystyle {\mathfrak {g}}} -valued 1-form, where g {\displaystyle {\mathfrak {g}}} is the Lie algebra of G.

• H. B. Nielsen and P. Olesen (1973). "Vortex-line models for dual strings". Nuclear Physics B. 61: 45–61. Bibcode:1973NuPhB..61...45N. doi:10.1016/0550-3213(73)90350-7.

• Peskin, M and Schroeder, D.; An Introduction to Quantum Field Theory (Westview Press, 1995) ISBN 0-201-50397-2