Final Recap

We use natural units $c=\hslash =1$ throughout.

Classical System

• Given the force laws and suitable initial position and velocity, the equations determine future and past motion wherever a unique solution exists.
• Newtonian Formulation
  • For a constant-mass particle in an inertial frame, $\mathbf{F}=m\mathbf{a}$.
  • Tracking forces and constraints can become cumbersome in complicated systems.
  • The vector law does not depend on a particular choice of Cartesian axes; its component expressions depend on the coordinates used.
• Lagrangian Formulation
  • For ordinary particles with standard kinetic energy and a velocity-independent potential, $L=K-V$. More general systems can require other forms of $L$.
  • Action: $S={\int }_{t_i}^{t_f}L(q,\dot{q},t)dt$.
  • Physical trajectories make the action stationary: $\delta S=0$. A stationary point need not be a minimum or maximum.
  • Stationary action gives the Euler–Lagrange equations: $\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=0$.
  • The usual variational derivation fixes the endpoint positions. These are boundary conditions for the variation, not initial conditions for predicting motion; we normally solve the resulting equations using initial positions and velocities.
  • We choose generalized coordinates and encode the dynamics in $L$, often avoiding explicit constraint forces.
• Hamiltonian Formulation
  • Define the canonical momenta $p_i=\partial L/\partial {\dot{q}}_i$. When these relations can be inverted for the velocities, use $(q_i,p_i)$ as phase-space coordinates.
  • The Hamiltonian is $H(q,p,t)={\sum }_i{p}_i{\dot{q}}_i-L$, with the velocities expressed in terms of $q,p,t$. For standard kinetic energy and a velocity-independent potential in time-independent coordinates, $H=K+V$; this equality is not universal.
  • For each degree of freedom, replace a second-order ordinary differential equation with two first-order ordinary differential equations: ${\dot{q}}_i=\partial H/\partial p_i$ and ${\dot{p}}_i=-\partial H/\partial q_i$.
  • The system follows a trajectory in $(q,p)$ phase space. Along that trajectory, $dH/dt=\partial H/\partial t$, so $H$ stays constant when it has no explicit time dependence.

Quantum System

• A quantum state generally predicts a distribution of measurement outcomes. Position and momentum cannot both have arbitrarily sharp distributions. An isolated state's evolution under the Schrödinger equation is deterministic, even though individual measurement outcomes are generally probabilistic.
• A pure state is represented by a normalized vector $|\varphi ⟩$, up to an overall phase. A single measurement does not reveal that entire vector; mixed states require density operators.
• Observables are self-adjoint linear operators (Hermitian matrices in finite dimensions), so their possible measured values are real.
• A linear operator maps vectors in its domain to vectors. Its output need not be normalized and can be zero, so it is not automatically another physical state.
• An eigenvector satisfies $\hat{A}|a_n⟩=a_n|a_n⟩$: the operator maps the nonzero vector to a scalar multiple of itself, and $a_n$ is its eigenvalue. General operators can have complex eigenvalues, but self-adjoint observables have real ones.
• For an observable with a discrete orthonormal eigenbasis, expand the state itself as $|\varphi ⟩={\sum }_n{c}_n|a_n⟩$, where $c_n=⟨a_n|\varphi ⟩$.
• For a nondegenerate eigenvalue $a_n$, the probability of measuring it is $|c_n{|}^2$. The coefficient $c_n$ is a probability amplitude. For a degenerate eigenvalue, sum $|c_n{|}^2$ over an orthonormal basis of its eigenspace.
• Applying $\hat{A}$ to a state is not a model of performing a measurement. In an ideal projective measurement, the observed outcome selects an eigenspace, and projection followed by normalization gives the conditional state after measurement.
• Useful examples of observables:
  • Position: possible outcomes are positions in space. In the position representation, $(\hat{x}\varphi )(x)=x\varphi (x)$.
  • Momentum: possible outcomes are momenta. In the position representation for a particle on a line, $(\hat{p}\varphi )(x)=-i{\partial }_x\varphi (x)$.
• The wavefunction $\varphi (x)=⟨x|\varphi ⟩$ expresses the state in the position basis. It is not the result of applying $\hat{x}$ to the state.
• The momentum-space wavefunction $\tilde{\varphi }(p)=⟨p|\varphi ⟩$ expresses the same state in a different basis. A Fourier transform relates these two wavefunctions.
• Position and momentum on an infinite line have continuous spectra and use generalized eigenstates. Here $|\varphi (x){|}^2$ is a probability density: the probability of finding the particle in a region $R$ is ${\int }_R|\varphi (x){|}^{2}dx$.
• Evolution of a quantum system
  • The initial state determines how an isolated system evolves, so we usually study its time dependence.
  • The Schrödinger equation governs the time evolution of the state: $i{\partial }_t|\varphi (t)⟩=\hat{H}|\varphi (t)⟩$.
• Harmonic Oscillator

  • Important model system and the basis for ladder-operator quantization

  • Restoring force proportional to displacement: $F=-kx$

  • Potential energy is quadratic about equilibrium: $V(x)=\frac{1}{2}k{x}^2$

  • Hamiltonian: $H(x,p)=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2$, where $k=m{\omega }^2$

  • Constant-energy trajectories are ellipses in ordinary $(x,p)$ phase space and circles after rescaling the axes

  • Quantization promotes $x$ and $p$ to operators satisfying $[\hat{x},\hat{p}]=i$. Define

    $$\hat{a}=\sqrt{\frac{m\omega }{2}}\hat{x}+\frac{i}{\sqrt{2m\omega }}\hat{p},{\hat{a}}^{†}=\sqrt{\frac{m\omega }{2}}\hat{x}-\frac{i}{\sqrt{2m\omega }}\hat{p},$$

    which obey $[\hat{a},{\hat{a}}^{†}]=1$.

  • With the number operator $\hat{N}={\hat{a}}^{†}\hat{a}$,

    $$\hat{H}=\omega \left(\hat{N}+\frac{1}{2}\right).$$

    The ground state satisfies $\hat{a}|0⟩=0$, and the normalized number states are $|n⟩=({\hat{a}}^{†}{)}^n|0⟩/\sqrt{n!}$.

  • The ladder operators act as

    $$\hat{a}|n⟩=\sqrt{n}|n-1⟩,{\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩,$$

    giving the discrete spectrum $E_n=\omega (n+\frac{1}{2})$. The ground state therefore has nonzero zero-point energy $E_0=\omega /2$.

• First Quantization
  • In canonical quantization, promote position and momentum to operators satisfying $[\hat{x},\hat{p}]=i$, then construct the Hamiltonian operator $\hat{H}$ from them.
  • Products involving noncommuting variables can introduce operator-ordering ambiguities.
• Special Relativity

  • Particle physics often involves particles moving near the speed of light, so relativistic effects must be included.

  • Lorentz transformations relate spacetime coordinates in different inertial frames with a common origin.

  • For a boost with relative velocity $v$ along the $x$ direction,

    $$t^{\prime }=\gamma (t-vx),x^{\prime }=\gamma (x-vt),y^{\prime }=y,z^{\prime }=z,\gamma =\frac{1}{\sqrt{1-v^2}}.$$

    The inverse boost is obtained by replacing $v\to -v$. Unlike a Galilean transformation, a Lorentz boost mixes space and time and preserves the speed of light.

  • The Minkowski metric with signature $(+,-,-,-)$ gives the invariant spacetime interval squared: $\Delta s^2=(\Delta t{)}^2-|\Delta \mathbf{x}{|}^2$.

  • Energy and three-momentum form the four-momentum $p^{\mu }=(E,\mathbf{p})$, which transforms as a four-vector under Lorentz transformations.

  • Under the same boost,

    $$E^{\prime }=\gamma (E-v{p}_x),p_x^{\prime }=\gamma (p_x-vE),p_y^{\prime }=p_y,p_z^{\prime }=p_z.$$

  • Its invariant squared norm is $p^{\mu }{p}_{\mu }=E^2-|\mathbf{p}{|}^2=m^2$, where $m$ is the rest mass.

• Klein–Gordon Equation
  • The Klein–Gordon equation follows from the relativistic energy–momentum relation $E^2=|\mathbf{p}{|}^2+m^2$ by substituting $E\to i{\partial }_t$ and $\mathbf{p}\to -i\nabla$, giving $({\partial }_t^2-{\nabla }^2+m^2)\varphi =0$.
  • Has positive- and negative-frequency plane-wave solutions, with $E=\pm \sqrt{|\mathbf{p}{|}^2+m^2}$. In quantum field theory, these are interpreted through particle and antiparticle modes with positive physical energies. For a real scalar field, the particle is its own antiparticle.
  • Is second order in time, so specifying a solution requires both $\varphi$ and ${\partial }_t\varphi$ initially.
  • Its conserved density is not positive definite, so it cannot serve as an ordinary position probability density. For a complex field, it is interpreted as a charge density.
  • Describes spin-0 fields; particles with other spins require different field equations.
• Plane-Wave Solutions
  • A free particle with definite momentum has a plane wave $\psi (t,\mathbf{x})=A{e}^{i(\mathbf{p}\cdot \mathbf{x}-Et)}=A{e}^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}$, where $\mathbf{p}=\mathbf{k}$ and $E=\omega$ in natural units.
  • Nonrelativistic: this solves the free Schrödinger equation $i{\partial }_t\psi =-\frac{1}{2m}{\nabla }^2\psi$ when the kinetic energy is $E=|\mathbf{p}{|}^2/(2m)$. The corresponding relativistic total energy has the low-momentum expansion $E_{\mathrm{rel}}\approx m+|\mathbf{p}{|}^2/(2m)$.
  • Relativistic: a free scalar field has plane-wave solutions $\varphi (x)=A{e}^{-i{p}_{\mu }{x}^{\mu }}$, where $x^{\mu }=(t,\mathbf{x})$ and $p_{\mu }{x}^{\mu }=Et-\mathbf{p}\cdot \mathbf{x}$ is Lorentz invariant. These solve the Klein–Gordon equation when $E^2=|\mathbf{p}{|}^2+m^2$.
  • A single plane wave extends throughout space and is not square-normalizable on infinite space; superpositions can form localized, normalizable wave packets.
• Free Dirac Equation
  • The Dirac equation factorizes the Klein–Gordon operator using gamma matrices, which satisfy the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }I$.
  • Is first order in time and space: $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$.
  • Applying $(i{\gamma }^{\mu }{\partial }_{\mu }+m)$ shows that the free Dirac equation implies $(\square +m^2)\psi =0$, where $\square ={\partial }_t^2-{\nabla }^2$. Thus each spinor component satisfies the Klein–Gordon equation, and plane-wave solutions satisfy $E^2=|\mathbf{p}{|}^2+m^2$. The Klein–Gordon equation alone does not imply the Dirac equation.
  • Has a nonnegative conserved probability density $\rho ={\psi }^{†}\psi$ in the single-particle interpretation.
  • Describes spin-$\frac{1}{2}$ particles.
  • Still has positive- and negative-energy solutions; quantum field theory interprets these through particles and antiparticles with positive physical energies.
  • General solution

    • Because the free Dirac equation is linear, its general solution is a superposition of all momentum, spin, and frequency modes:

      $$\psi (x)=\sum _{s=1}^2\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\left[a_s(\mathbf{p})u_s(\mathbf{p})e^{-ip\cdot x}+b_s^{\ast }(\mathbf{p})v_s(\mathbf{p})e^{+ip\cdot x}\right],E_{\mathbf{p}}=\sqrt{|\mathbf{p}{|}^2+m^2}.$$

    • Here $p\cdot x=E_{\mathbf{p}}t-\mathbf{p}\cdot \mathbf{x}$. The $u_s$ terms are positive-frequency modes and the $v_s$ terms are negative-frequency modes, with two spin labels $s=1,2$ for each momentum.

    • For a classical Dirac field, $a_s(\mathbf{p})$ and $b_s^{\ast }(\mathbf{p})$ are numerical amplitudes fixed by the initial field. In quantum field theory they become electron-annihilation and positron-creation operators, respectively.

    • The field is therefore a superposition of independent momentum–spin modes. After quantization, each mode behaves as a fermionic oscillator, governed by anticommutation relations rather than the commutation relations of an ordinary bosonic harmonic oscillator.

  • Spin
    • Spin is intrinsic angular momentum associated with how the Dirac spinor transforms under rotations; it is not literal rotation of a particle. Angular momentum conservation reveals why orbital angular momentum alone is insufficient.
    • For the free Dirac Hamiltonian $H=\mathbf{\alpha }\cdot \mathbf{p}+\beta m$, with ${\alpha }^i={\gamma }^0{\gamma }^i$ and $\beta ={\gamma }^0$, orbital angular momentum $\mathbf{L}=\mathbf{x}\times \mathbf{p}$ is not separately conserved: $[H,L_i]\ne 0$ in general.
    • Spin operator: $\mathbf{S}=\frac{1}{2}\mathbf{\Sigma }$, where in the Dirac representation ${\Sigma }_i=\left(\begin{array}{cc}{\sigma }_i& 0\\ 0& {\sigma }_i\end{array}\right)$ and ${\sigma }_i$ are the Pauli matrices. It obeys $[S_i,S_j]=i{ϵ}_{ijk}{S}_k$ and ${\mathbf{S}}^2=\frac{3}{4}I$, corresponding to spin $s=\frac{1}{2}$.
    • Total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S}$ is conserved: $[H,J_i]=0$. More precisely, the two commutators cancel, $[H,L_i]+[H,S_i]=0$, although neither $\mathbf{L}$ nor $\mathbf{S}$ is separately conserved in general.
    • Along the $z$ axis, the two-component spin basis is ${\chi }_{+}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and ${\chi }_{-}=\left(\begin{array}{c}0\\ 1\end{array}\right)$, with $({\sigma }_z/2){\chi }_{\pm }=\pm \frac{1}{2}{\chi }_{\pm }$.
    • Positive-frequency spinor solutions have the form ${\psi }_s(x)=u_s(\mathbf{p})e^{-i{p}_{\mu }{x}^{\mu }}$, where $E=\sqrt{|\mathbf{p}{|}^2+m^2}$ and $({\gamma }^{\mu }{p}_{\mu }-m)u_s=0$.
    • For $m>0$, in the Dirac representation one choice is $u_s(\mathbf{p})=\sqrt{E+m}\left(\begin{array}{c}{\chi }_s\\ \frac{\mathbf{\sigma }\cdot \mathbf{p}}{E+m}{\chi }_s\end{array}\right)$, normalized so that $u_s^{†}{u}_s=2E$. The label $s=\pm$ specifies the rest-frame spin projection; a moving spinor need not be an eigenvector of $S_z$.
    • Negative-frequency solutions are ${\psi }_s(x)=v_s(\mathbf{p})e^{+i{p}_{\mu }{x}^{\mu }}$ with $E>0$ and $({\gamma }^{\mu }{p}_{\mu }+m)v_s=0$. In quantum field theory, these modes enter the antiparticle part of the field.
    • A Dirac spinor has four components, but the free equation leaves two independent spin states for each energy branch. Under a $2\pi$ spatial rotation, a spinor changes sign; it returns to itself after $4\pi$.

Classical Field to Quantum Field

• A classical field assigns a number, vector, or spinor to every spacetime point. Examples include a scalar field $\varphi (x)$, the electromagnetic field $A^{\mu }(x)$, and a Dirac spinor $\psi (x)$.

• Its dynamics follow from an action $S=\int d^{4}x\mathcal{L}$. Stationary action gives the field Euler–Lagrange equation

  $${\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\varphi )}\right)-\frac{\partial \mathcal{L}}{\partial \varphi }=0.$$

• A free field obeys a linear equation, so it can be decomposed into independent Fourier modes. For a free scalar field, each momentum mode satisfies

  $${\ddot{\varphi }}_{\mathbf{p}}(t)+E_{\mathbf{p}}^2{\varphi }_{\mathbf{p}}(t)=0,E_{\mathbf{p}}=\sqrt{|\mathbf{p}{|}^2+m^2},$$

  which has the same form as a harmonic oscillator with frequency $E_{\mathbf{p}}$.

• Canonical field quantization promotes the classical field $\varphi (t,\mathbf{x})$ and its conjugate momentum $\pi (t,\mathbf{x})=\partial \mathcal{L}/\partial \dot{\varphi }$ to operators with the equal-time relation

  $$[\hat{\varphi }(t,\mathbf{x}),\hat{\pi }(t,\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y}).$$

• Equivalently, promote each mode amplitude to creation and annihilation operators. A free real scalar field becomes

  $$\hat{\varphi }(x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\left({\hat{a}}_{\mathbf{p}}{e}^{-ip\cdot x}+{\hat{a}}_{\mathbf{p}}^{†}{e}^{+ip\cdot x}\right),$$

  with $[{\hat{a}}_{\mathbf{p}},{\hat{a}}_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$.

• The vacuum $|0⟩$ satisfies ${\hat{a}}_{\mathbf{p}}|0⟩=0$. Acting with ${\hat{a}}_{\mathbf{p}}^{†}$ creates one particle of momentum $\mathbf{p}$ and energy $E_{\mathbf{p}}$; repeated creation operators build multiparticle Fock states.

• The free-field Hamiltonian is a sum of oscillator Hamiltonians. In a finite box with discrete momenta, it has the schematic form $\hat{H}={\sum }_{\mathbf{p}}{E}_{\mathbf{p}}({\hat{N}}_{\mathbf{p}}+\frac{1}{2})$. The $\frac{1}{2}{E}_{\mathbf{p}}$ terms make up the vacuum energy, and each excitation above the vacuum is one particle quantum of a field mode.

• Bosonic fields use commutators and permit any occupation number. Dirac fields use anticommutators, so each momentum–spin particle or antiparticle mode can have occupation number only $0$ or $1$.

• In a free theory the momentum modes evolve independently. Interactions couple the fields and allow quanta to be created, destroyed, or scattered while conserving the symmetries' associated quantities.

Free Lagrangian Densities

• Real scalar field — Klein–Gordon

  $${\mathcal{L}}_{\mathrm{KG}}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2.$$

  The Euler–Lagrange equation is $(\square +m^2)\varphi =0$. After quantization, the field describes neutral spin-$0$ bosons; a real scalar particle is its own antiparticle. A complex scalar field instead uses $\mathcal{L}={\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi$ and can carry a conserved charge.

• Dirac field

  $${\mathcal{L}}_{\mathrm{D}}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi ,\bar{\psi }={\psi }^{†}{\gamma }^0.$$

  Varying $\bar{\psi }$ gives $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$; varying $\psi$ gives the adjoint Dirac equation. After quantization, $\psi$ describes spin-$\frac{1}{2}$ fermions and $\bar{\psi }$ is its Dirac adjoint. Particle and antiparticle modes obey anticommutation relations.

• Electromagnetic field — Maxwell

  $${\mathcal{L}}_{\mathrm{M}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu },F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.$$

  The Euler–Lagrange equation is the source-free Maxwell equation ${\partial }_{\mu }{F}^{\mu \nu }=0$. The Lagrangian is invariant under the gauge transformation $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\Lambda$. After quantization, the field describes massless spin-$1$ photons with two physical transverse polarizations.

• Adding an external conserved current gives the interaction term ${\mathcal{L}}_{\mathrm{int}}=-j_{\mu }{A}^{\mu }$ and changes Maxwell's equation to ${\partial }_{\mu }{F}^{\mu \nu }=j^{\nu }$.

• Each free Lagrangian is quadratic in its field, so its equation of motion is linear and its momentum modes evolve independently. Interaction terms contain products of fields and couple those modes.

Fermion and Photon Quantization

• Quantizing the Dirac field promotes its mode amplitudes to operators:

  $$\hat{\psi }(x)=\sum _{s=1}^2\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\left[{\hat{a}}_s(\mathbf{p})u_s(\mathbf{p})e^{-ip\cdot x}+{\hat{b}}_s^{†}(\mathbf{p})v_s(\mathbf{p})e^{+ip\cdot x}\right].$$

  Here ${\hat{a}}_s$ annihilates a fermion and ${\hat{b}}_s^{†}$ creates its antifermion.

• Fermionic modes obey canonical anticommutation relations:

  $$\{{\hat{a}}_s(\mathbf{p}),{\hat{a}}_{s^{\prime }}^{†}(\mathbf{q})\}=(2\pi {)}^3{\delta }_{s{s}^{\prime }}{\delta }^3(\mathbf{p}-\mathbf{q}),$$

  $$\{{\hat{b}}_s(\mathbf{p}),{\hat{b}}_{s^{\prime }}^{†}(\mathbf{q})\}=(2\pi {)}^3{\delta }_{s{s}^{\prime }}{\delta }^3(\mathbf{p}-\mathbf{q}),$$

  with all other anticommutators zero. Consequently, $({\hat{a}}_s^{†}{)}^2=({\hat{b}}_s^{†}{)}^2=0$ for a fixed mode, implementing the Pauli exclusion principle and restricting each mode's occupation to $0$ or $1$.

• The vacuum satisfies ${\hat{a}}_s(\mathbf{p})|0⟩={\hat{b}}_s(\mathbf{p})|0⟩=0$. The operators ${\hat{a}}_s^{†}$ and ${\hat{b}}_s^{†}$ create particle and antiparticle states. After normal ordering, the free Hamiltonian is a sum of positive-energy particle and antiparticle number operators.

• The Maxwell field has gauge redundancy, so its four potential components do not represent four physical photon polarizations. For external physical photons, one may use the transverse-mode expansion

  $${\hat{A}}_{\mu }(x)=\sum _{\lambda =1}^2\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2|\mathbf{p}|}}\left[{\hat{a}}_{\lambda }(\mathbf{p}){ϵ}_{\mu }^{(\lambda )}(\mathbf{p})e^{-ip\cdot x}+{\hat{a}}_{\lambda }^{†}(\mathbf{p}){ϵ}_{\mu }^{(\lambda )\ast }(\mathbf{p})e^{+ip\cdot x}\right],$$

  where $p^0=|\mathbf{p}|$ and $\lambda =1,2$ labels the two physical polarizations.

• Photon operators obey bosonic commutation relations,

  $$[{\hat{a}}_{\lambda }(\mathbf{p}),{\hat{a}}_{{\lambda }^{\prime }}^{†}(\mathbf{q})]=(2\pi {)}^3{\delta }_{\lambda {\lambda }^{\prime }}{\delta }^3(\mathbf{p}-\mathbf{q}),$$

  so any number of photons may occupy the same mode.

• Covariant perturbation theory adds the gauge-fixing term

  $${\mathcal{L}}_{\mathrm{gf}}=-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2.$$

  In Feynman gauge, $\xi =1$, the internal photon propagator is $-i{g}_{\mu \nu }/(p^2+iϵ)$. Four formal components appear internally, but gauge invariance ensures that only the two physical transverse polarizations contribute to observable external photon states.

• The rest of the recap builds the Standard Model Lagrangian from matter- and gauge-field kinetic terms, gauge interactions, the Higgs sector, and Yukawa interactions. After electroweak symmetry breaking, the Higgs and Yukawa terms generate masses for the massive gauge bosons and fermions.

Electromagnetic Field and QED

• The free Dirac and Maxwell Lagrangians describe electrons and photons propagating independently. Quantum electrodynamics (QED) adds the interaction between them.

• Global $U(1)$ symmetry: the free Dirac Lagrangian is invariant under a constant phase rotation

  $$\psi \to e^{i\alpha }\psi ,\bar{\psi }\to e^{-i\alpha }\bar{\psi }.$$

  Noether's theorem associates this symmetry with the conserved electric current $j^{\mu }=q\bar{\psi }{\gamma }^{\mu }\psi$, satisfying ${\partial }_{\mu }{j}^{\mu }=0$. Its conserved charge is $Q=\int d^{3}x{j}^0$.

• Local $U(1)$ symmetry: if the phase becomes position dependent, $\alpha \to \alpha (x)$, differentiating $e^{i\alpha (x)}\psi$ produces an extra term, so the ordinary derivative does not transform covariantly.

• Introduce the electromagnetic gauge field $A_{\mu }$ and the covariant derivative

  $$D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }.$$

  Under

  $$\psi \to e^{i\alpha (x)}\psi ,A_{\mu }\to A_{\mu }-\frac{1}{q}{\partial }_{\mu }\alpha ,$$

  the covariant derivative transforms as $D_{\mu }\psi \to e^{i\alpha (x)}{D}_{\mu }\psi$. The gauge field compensates for the position-dependent phase.

• The field strength

  $$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$$

  is gauge invariant. The term $-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$ supplies the gauge field's dynamics and gives Maxwell's equations.

• The complete QED Lagrangian is

  $${\mathcal{L}}_{\mathrm{QED}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\bar{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi .$$

  Expanding $D_{\mu }$ separates it into

  $${\mathcal{L}}_{\mathrm{QED}}=\underset{\text{free photon}}{\underbrace{-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }}}+\underset{\text{free electron/positron}}{\underbrace{\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi }}\underset{\text{interaction}}{\underbrace{-q\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }}}.$$

• The interaction is ${\mathcal{L}}_{\mathrm{int}}=-j^{\mu }{A}_{\mu }$. It allows a charged fermion to emit or absorb a photon and gives the QED vertex factor $-iq{\gamma }^{\mu }$. Charge and four-momentum are conserved at each vertex.

• Because $U(1)$ is Abelian, $F_{\mu \nu }$ contains no terms quadratic in $A_{\mu }$; therefore QED has no elementary photon-only interaction vertex. Photons can still scatter indirectly through charged-particle loops.

• A photon mass term $m_{\gamma }^2{A}_{\mu }{A}^{\mu }/2$ is not gauge invariant, so unbroken electromagnetic $U(1)$ requires the photon to be massless. Quantizing $A_{\mu }$ leaves two physical transverse photon polarizations.

Weak Interaction

• Experiments such as beta decay show that the weak interaction violates parity, which reverses spatial coordinates and momenta: $\mathbf{x}\to -\mathbf{x}$ and $\mathbf{p}\to -\mathbf{p}$. Spin is unchanged under parity, so the correlation $\mathbf{S}\cdot \mathbf{p}$ changes sign. The observed asymmetry means the weak interaction distinguishes left from right.

• Define

  $${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3,P_L=\frac{1-{\gamma }^5}{2},P_R=\frac{1+{\gamma }^5}{2}.$$

  These chiral projectors satisfy $P_L^2=P_L$, $P_R^2=P_R$, $P_L{P}_R=0$, and $P_L+P_R=1$. They split a Dirac field into ${\psi }_L=P_L\psi$ and ${\psi }_R=P_R\psi$.

• Chirality is a Lorentz-transformation property of a spinor and is distinct from spin projection. For massless particles it agrees with helicity; for massive particles, a state of definite helicity generally contains both chiralities.

• The charged weak interaction couples only to left-chiral fermion fields. A left-chiral neutrino field annihilates negative-helicity neutrinos and creates positive-helicity antineutrinos in the massless limit. There is no corresponding right-chiral charged current.

• For the electron and electron neutrino, the left-chiral current and its conjugate are

  $$j_{\ell }^{\mu }={\bar{e}}_L{\gamma }^{\mu }{\nu }_{eL}=\bar{e}{\gamma }^{\mu }{P}_L{\nu }_e,(j_{\ell }^{\mu }{)}^{†}={\bar{\nu }}_{eL}{\gamma }^{\mu }{e}_L.$$

  The analogous quark current ${\bar{u}}_L{\gamma }^{\mu }{d}_L$ changes a down quark into an up quark. These currents account for the species changes in beta decay.

• Group the left-chiral fields into weak-isospin doublets

  $$L=\left(\begin{array}{c}{\nu }_{eL}\\ e_L\end{array}\right),Q=\left(\begin{array}{c}{u}_L\\ d_L\end{array}\right).$$

  A global $SU(2)$ transformation acts as $L\to UL$, where $U=\exp (i{\alpha }^a{T}^a)$ and $T^a={\sigma }^a/2$. The Pauli matrices here mix the two species in a doublet; they do not act on spin.

• Making the symmetry local, $U\to U(x)$, requires one gauge field for each of the three $SU(2)$ generators. Define

  $$D_{\mu }={\partial }_{\mu }+ig{W}_{\mu }^a{T}^a,$$

  where $g$ is the weak coupling and $W_{\mu }^1,W_{\mu }^2,W_{\mu }^3$ are the three gauge fields. Because this symmetry acts on left-chiral doublets, it is called $SU(2{)}_L$; right-chiral charged fermions are $SU(2{)}_L$ singlets.

• The combinations

  $$W_{\mu }^{\pm }=\frac{W_{\mu }^1\mp i{W}_{\mu }^2}{\sqrt{2}}$$

  are the charged weak fields. The third field $W_{\mu }^3$ is neutral. Before electroweak mixing, $W^3$ is not yet the physical $Z$ boson or photon.

• Expanding the covariant derivative gives the lepton interaction

  $$\begin{array}{rl}{\mathcal{L}}_{\mathrm{int}}=& -\frac{g}{\sqrt{2}}\left({\bar{\nu }}_{eL}{\gamma }^{\mu }{e}_L{W}_{\mu }^{+}+{\bar{e}}_L{\gamma }^{\mu }{\nu }_{eL}{W}_{\mu }^{-}\right)\\ & -\frac{g}{2}\left({\bar{\nu }}_{eL}{\gamma }^{\mu }{\nu }_{eL}-{\bar{e}}_L{\gamma }^{\mu }{e}_L\right)W_{\mu }^3.\end{array}$$

  The first line contains the charged-current interactions, which exchange an electron and neutrino through $W^{\pm }$. The second line is the neutral $SU(2{)}_L$ interaction; electroweak mixing later turns the neutral gauge fields into the physical $Z$ and photon.

• The $SU(2)$ field strength and gauge-field kinetic term are

  $$W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a-g{ϵ}^{abc}{W}_{\mu }^b{W}_{\nu }^c,{\mathcal{L}}_{\mathrm{YM}}=-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }.$$

  The extra product of gauge fields appears because $SU(2)$ is non-Abelian. It produces direct three- and four-gauge-boson interactions, unlike the Abelian photon field in QED.

• Local $SU(2{)}_L$ symmetry forbids an explicit mass term for the gauge fields. In the full electroweak theory, the Higgs mechanism gives masses to $W^{\pm }$ and $Z$ while leaving the photon massless.

Electromagnetic and Weak Terms So Far

• Collect the first-generation fermions as $f\in \{{\nu }_e,e,u,d\}$ and the left-chiral doublets as

  $$L=\left(\begin{array}{c}{\nu }_{eL}\\ e_L\end{array}\right),Q=\left(\begin{array}{c}{u}_L\\ d_L\end{array}\right).$$

  The electromagnetic current is

  $$J_{\mathrm{EM}}^{\mu }=\sum _f{q}_f\bar{f}{\gamma }^{\mu }f,$$

  where $q_f$ is the electric charge of each field.

• A compact bookkeeping summary of the free gauge fields, free fermions, and interactions introduced so far is

  $$\boxed{\begin{array}{rl}{\mathcal{L}}_{\mathrm{so}\mathrm{far}}=& -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }+\sum _f\bar{f}(i{\gamma }^{\mu }{\partial }_{\mu }-m_f)f\\ & -J_{\mathrm{EM}}^{\mu }{A}_{\mu }-g\bar{L}{\gamma }^{\mu }{T}^{a}L{W}_{\mu }^a-g\bar{Q}{\gamma }^{\mu }{T}^{a}Q{W}_{\mu }^a.\end{array}}$$

• The terms have the following roles:

  • $-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$: photon propagation.
  • $-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }$: weak gauge-boson propagation and self-interactions.
  • ${\sum }_f\bar{f}(i{\gamma }^{\mu }{\partial }_{\mu }-m_f)f$: free fermion propagation in the broken-phase, low-energy description.
  • $-J_{\mathrm{EM}}^{\mu }{A}_{\mu }$: photon coupling to every electrically charged fermion.
  • $-g\bar{L}{\gamma }^{\mu }{T}^{a}L{W}_{\mu }^a-g\bar{Q}{\gamma }^{\mu }{T}^{a}Q{W}_{\mu }^a$: weak couplings to left-chiral lepton and quark doublets.

• Rewriting $W^1$ and $W^2$ as $W^{\pm }$ exposes the charged-current part

  $${\mathcal{L}}_{\mathrm{CC}}=-\frac{g}{\sqrt{2}}\left({\bar{\nu }}_{eL}{\gamma }^{\mu }{e}_L{W}_{\mu }^{+}+{\bar{e}}_L{\gamma }^{\mu }{\nu }_{eL}{W}_{\mu }^{-}+{\bar{u}}_L{\gamma }^{\mu }{d}_L{W}_{\mu }^{+}+{\bar{d}}_L{\gamma }^{\mu }{u}_L{W}_{\mu }^{-}\right).$$

• This is an intermediate bookkeeping expression, rather than the final gauge-invariant electroweak Lagrangian. In particular, the displayed fermion masses belong to the broken-phase, low-energy description. Before symmetry breaking, electromagnetism and the weak interaction are organized as $SU(2{)}_L\times U(1{)}_Y$, with a hypercharge field $B_{\mu }$ instead of a separate photon field and with fermion masses replaced by Yukawa couplings to the Higgs. The Higgs mechanism later mixes $W_{\mu }^3$ and $B_{\mu }$ into the physical photon $A_{\mu }$ and $Z_{\mu }$, and generates the fermion and massive gauge-boson mass terms.

Electroweak Unification

• The neutral $SU(2{)}_L$ field $W_{\mu }^3$ cannot be the photon because its couplings have the wrong pattern:

  Fermion component	$t_3$	Coupling to $W^3$	Required photon coupling
  ${\nu }_{eL}$	$+\frac{1}{2}$	nonzero	$0$
  $e_L$	$-\frac{1}{2}$	nonzero	$-e$
  $e_R$	$0$	$0$	$-e$

  The photon must couple equally to $e_L$ and $e_R$ because they are two chiral components of the same charged electron, while it must not couple to the neutrino. The $W^3$ field instead couples only according to weak isospin $t_3$.

• Add a second symmetry, weak hypercharge $U(1{)}_Y$, with generator $Y/2$. Electric charge is then

  $$\boxed{Q=T^3+\frac{Y}{2}}.$$

  Hypercharge supplies the common offset that $T^3$ alone lacks. In the convention $Q=T^3+Y/2$, the minimal Standard Model assignments for each generation and the Higgs are

  Multiplet	$SU(3{)}_C$	$SU(2{)}_L$	$t_3$	$Y$	$Q$
  $L=({\nu }_L,e_L)$	$1$	$2$	$(+\frac{1}{2},-\frac{1}{2})$	$-1$	$(0,-1)$
  $e_R$	$1$	$1$	$0$	$-2$	$-1$
  $Q_L=(u_L,d_L)$	$3$	$2$	$(+\frac{1}{2},-\frac{1}{2})$	$+\frac{1}{3}$	$(+\frac{2}{3},-\frac{1}{3})$
  $u_R$	$3$	$1$	$0$	$+\frac{4}{3}$	$+\frac{2}{3}$
  $d_R$	$3$	$1$	$0$	$-\frac{2}{3}$	$-\frac{1}{3}$
  $\Phi =({\varphi }^{+},{\varphi }^0)$	$1$	$2$	$(+\frac{1}{2},-\frac{1}{2})$	$+1$	$(+1,0)$

  The same gauge assignments repeat for all three generations. The minimal model contains no right-chiral neutrino; adding ${\nu }_R$ for a Dirac neutrino mass would make it a complete gauge singlet with $Y=Q=0$.

• Making $U(1{)}_Y$ local introduces the hypercharge gauge field $B_{\mu }$ with coupling $g^{\prime }$. For a field $F$ with weak isospin generators $T^a$ and hypercharge $Y_F$,

  $$D_{\mu }F=\left({\partial }_{\mu }+ig{T}^a{W}_{\mu }^a+i{g}^{\prime }\frac{Y_F}{2}{B}_{\mu }\right)F.$$

  The electroweak gauge group is therefore $SU(2{)}_L\times U(1{)}_Y$.

• Neither $W_{\mu }^3$ nor $B_{\mu }$ alone has the correct photon coupling. Define the weak mixing angle ${\theta }_W$, with $s_W=\sin {\theta }_W$ and $c_W=\cos {\theta }_W$, and rotate the two neutral fields:

  $$A_{\mu }=s_W{W}_{\mu }^3+c_W{B}_{\mu },Z_{\mu }=c_W{W}_{\mu }^3-s_W{B}_{\mu }.$$

  The couplings obey

  $$e=g{s}_W=g^{\prime }{c}_W,$$

  so the $A_{\mu }$ coefficient becomes $e(t_3+Y/2)=eQ$. Thus $A_{\mu }$ couples to electric charge with the universal interaction $-e{Q}_f\bar{f}{\gamma }^{\mu }f{A}_{\mu }$.

• The orthogonal field $Z_{\mu }$ couples to the neutral weak current:

  $${\mathcal{L}}_Z=-\frac{g}{c_W}\left(t_3-s_W^2{Q}_f\right)\bar{f}{\gamma }^{\mu }f{Z}_{\mu }$$

  for a chiral component with quantum numbers $t_3$ and $Q_f$. Unlike the photon, the $Z$ generally couples differently to left- and right-chiral fermions.

• Electroweak unification reorganizes four gauge fields without changing their number:

  $$W^1,W^2,W^3,B⟶W^{+},W^{-},Z,A.$$

  The Higgs mechanism later makes $W^{\pm }$ and $Z$ massive while leaving the photon $A$ massless.

Electroweak Lagrangian So Far

• Before adding the Higgs field, the gauge-invariant electroweak theory contains the $SU(2{)}_L$ fields $W_{\mu }^a$, the hypercharge field $B_{\mu }$, and the chiral fermion multiplets. For one generation,

  $$F\in \{L,Q,e_R,u_R,d_R\},L=\left(\begin{array}{c}{\nu }_{eL}\\ e_L\end{array}\right),Q=\left(\begin{array}{c}{u}_L\\ d_L\end{array}\right).$$

• The electroweak gauge and fermion Lagrangian is

  $$\boxed{{\mathcal{L}}_{\mathrm{EW}}^{\mathrm{gauge}+\mathrm{fermion}}=-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }+\sum _{F}i\bar{F}{\gamma }^{\mu }{D}_{\mu }F}$$

  with

  $$D_{\mu }F=\left({\partial }_{\mu }+ig{T}^a{W}_{\mu }^a+i{g}^{\prime }\frac{Y_F}{2}{B}_{\mu }\right)F,$$

  $$W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a-g{ϵ}^{abc}{W}_{\mu }^b{W}_{\nu }^c,B_{\mu \nu }={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }.$$

  For right-chiral singlets, $T^a=0$, so only the hypercharge part of $D_{\mu }$ acts.

• After defining $W^{\pm }$ and rotating $W^3,B$ into $A,Z$, the fermion interactions can be collected as

  $$\boxed{{\mathcal{L}}_{\mathrm{int}}=-e{J}_{\mathrm{EM}}^{\mu }{A}_{\mu }-\frac{g}{\sqrt{2}}\left(J_{+}^{\mu }{W}_{\mu }^{+}+J_{-}^{\mu }{W}_{\mu }^{-}\right)-\frac{g}{c_W}{J}_Z^{\mu }{Z}_{\mu }}$$

  where, for the first generation,

  $$J_{\mathrm{EM}}^{\mu }=\sum _f{Q}_f\bar{f}{\gamma }^{\mu }f,$$

  $$J_{+}^{\mu }={\bar{\nu }}_{eL}{\gamma }^{\mu }{e}_L+{\bar{u}}_L{\gamma }^{\mu }{d}_L,J_{-}^{\mu }={\bar{e}}_L{\gamma }^{\mu }{\nu }_{eL}+{\bar{d}}_L{\gamma }^{\mu }{u}_L,$$

  $$J_Z^{\mu }=\sum _f\left(t_{3f}-s_W^2{Q}_f\right)\bar{f}{\gamma }^{\mu }f.$$

  The sum in $J_Z^{\mu }$ runs over chiral fermion components; $t_{3f}=0$ for right-chiral singlets.

• The three interaction terms describe:

  • Electromagnetism: the photon couples universally to electric charge $Q_f$.
  • Charged weak interaction: $W^{\pm }$ couples only to left-chiral doublets and changes one doublet member into the other.
  • Neutral weak interaction: $Z$ couples to both electric charge and weak isospin, so its left- and right-chiral couplings differ.

• The gauge kinetic term $-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }$ also produces $WW\gamma$, $WWZ$, and four-gauge-boson interactions after the field rotation. These follow from the non-Abelian $SU(2{)}_L$ field strength.

• This is the electroweak Lagrangian before the Higgs and Yukawa sectors. Explicit fermion, $W$, and $Z$ mass terms are absent because they would violate $SU(2{)}_L\times U(1{)}_Y$ gauge invariance. The Higgs field supplies those masses through spontaneous symmetry breaking while preserving a massless photon.

Higgs Mechanism

• The Higgs mechanism allows gauge bosons and fermions to acquire mass while the underlying Lagrangian remains gauge invariant.

Spontaneous Symmetry Breaking: Global $U(1)$ Example

• Begin with a complex scalar field and a symmetric potential

  $$\mathcal{L}={\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -V(\varphi ),V(\varphi )=-{\mu }^2{\varphi }^{\ast }\varphi +\lambda ({\varphi }^{\ast }\varphi {)}^2,{\mu }^2,\lambda >0.$$

  The Lagrangian is invariant under the global transformation $\varphi \to e^{i\alpha }\varphi$.

• The origin is unstable. Minimizing the potential gives a circle of degenerate vacua,

  $${\varphi }^{\ast }\varphi =\frac{{\mu }^2}{2\lambda }=\frac{v^2}{2},v^2=\frac{{\mu }^2}{\lambda }.$$

  Choose one vacuum, for example $⟨\varphi ⟩=v/\sqrt{2}$. The laws retain the $U(1)$ symmetry, but the chosen vacuum does not. This is spontaneous symmetry breaking.

• Expand around the chosen vacuum:

  $$\varphi (x)=\frac{1}{\sqrt{2}}\left(v+h(x)+i\chi (x)\right).$$

  The radial fluctuation $h$ is massive, while the angular fluctuation $\chi$ is a massless Goldstone boson:

  $$m_h^2=2\lambda v^2,m_{\chi }^2=0.$$

  A broken continuous global generator produces a massless Goldstone mode.

Local $U(1)$ Higgs Mechanism

• Now make the symmetry local by introducing a gauge field $A_{\mu }$:

  $$D_{\mu }={\partial }_{\mu }+iq{A}_{\mu },\mathcal{L}=(D_{\mu }\varphi {)}^{\ast }(D^{\mu }\varphi )-V(\varphi )-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }.$$

  This is a toy $U(1)$ gauge field used to demonstrate the mechanism; it is not the physical photon of the electroweak theory.

• Write the scalar in polar form,

  $$\varphi (x)=\frac{v+h(x)}{\sqrt{2}}{e}^{i\theta (x)/v}.$$

  A local gauge transformation can set $\theta (x)=0$. In unitary gauge,

  $$\varphi (x)=\frac{v+h(x)}{\sqrt{2}}.$$

  Gauge fixing removes the redundant phase from the scalar-field description; it does not discard a physical degree of freedom.

• Substitution into the scalar kinetic term gives

  $$(D_{\mu }\varphi {)}^{\ast }(D^{\mu }\varphi )=\frac{1}{2}({\partial }_{\mu }h{)}^2+\frac{1}{2}{q}^2(v+h{)}^2{A}_{\mu }{A}^{\mu }.$$

  The term $\frac{1}{2}{q}^2{v}^2{A}_{\mu }{A}^{\mu }$ is a vector mass term, so

  $$m_A=qv.$$

  The Goldstone degree of freedom becomes the longitudinal polarization of the massive vector. The counting is preserved: a massless vector has two polarizations and the complex scalar has two real modes; afterward, the massive vector has three polarizations and one real scalar $h$ remains.

Proca Equation After Higgs Generates a Vector Mass

• Keeping the free vector terms in unitary gauge gives

  $${\mathcal{L}}_{\mathrm{free}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{m}_A^2{A}_{\mu }{A}^{\mu },m_A=qv.$$

  The Maxwell derivative is $-F^{\mu \nu }$; differentiating the generated mass term gives $m_A^2{A}^{\nu }$. The Euler–Lagrange equation therefore gives the Proca equation,

  $$\boxed{{\partial }_{\mu }{F}^{\mu \nu }+m_A^2{A}^{\nu }=0.}$$

• Taking the divergence eliminates the antisymmetric field-strength term. For $m_A\ne 0$, the equation forces ${\partial }_{\mu }{A}^{\mu }=0$. Substituting this constraint back gives

  $$(\square +m_A^2)A^{\nu }=0,p^2=m_A^2,p\cdot \epsilon =0.$$

  In the rest frame, ${\epsilon }^0=0$ leaves three spatial polarizations. The longitudinal mode is physical. The free massive equation forces the constraint; we do not choose it as a gauge condition.

• Higgs explains the origin of the mass and longitudinal mode; Proca describes the resulting free massive vector. After electroweak symmetry breaking, the free $W^{\pm }$ and $Z$ fields have Proca equations, while the photon remains massless and obeys Maxwell. Interactions add terms to these free equations. See the derivation in the Higgs chapter.

Standard Model Higgs Field

• The electroweak Higgs field is a complex $SU(2{)}_L$ doublet with hypercharge $Y=1$:

  $$\Phi =\left(\begin{array}{c}{\varphi }^{+}\\ {\varphi }^0\end{array}\right),V(\Phi )=-{\mu }^2{\Phi }^{†}\Phi +\lambda ({\Phi }^{†}\Phi {)}^2.$$

• Choose the vacuum and unitary-gauge field as

  $$⟨\Phi ⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ v\end{array}\right),\Phi (x)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ v+h(x)\end{array}\right).$$

  The vacuum is electrically neutral because $Q⟨\Phi ⟩=(T^3+Y/2)⟨\Phi ⟩=0$. Thus $SU(2{)}_L\times U(1{)}_Y$ is reduced to the unbroken electromagnetic $U(1{)}_{\mathrm{EM}}$.

• The doublet contains four real degrees of freedom. Three Goldstone modes supply the longitudinal polarizations of $W^{+}$, $W^{-}$, and $Z$. The remaining radial excitation $h$ is the physical Higgs boson, with

  $$m_h^2=2\lambda v^2.$$