Scalar field theory ($\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V(\varphi )$)

A scalar field theory describes one or more Lorentz scalar fields through an action. The displayed Lagrangian density is a common classical normalization for a real scalar field; a complex scalar field has two real components. Canonical quantization turns these classical degrees of freedom into operators. The free real scalar field gives spin-zero particles, while interactions arise from the potential $V$.

A free real scalar field obeys the Klein-Gordon equation and creates neutral spin-zero particles.

A real scalar triplet groups three real scalar components into a vector under SO(3). Equivalently it is the three-dimensional Adjoint representation of SU(2), whose center acts trivially. Its rotationally invariant quadratic and quartic potentials depend on $\Phi \cdot \Phi$. A nonzero vacuum direction leaves rotations around that direction unbroken.

A relativistic real scalar field theory with a periodic cosine potential. In physical coordinates $y^{\mu }$, one normalization is $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi -m^2(1-\cos \beta \phi )/{\beta }^2$. With $\varphi =\beta \phi$ and $x^{\mu }=m{y}^{\mu }$, its action is $S={\beta }^{-2}\int d^{2}x[\frac{1}{2}(\partial \varphi {)}^2+\cos \varphi -1]$. Its Euler-Lagrange field equation is the Sine-Gordon equation. Distinct scalar-field vacua differ by $2\pi$ in $\varphi$, permitting a Sine-Gordon kink.

At renormalized coupling $\gamma =8\pi /n$, the kink-antikink transmission poles lie at $\vartheta =i\pi (1-k/n)$. The relativistic bound-state mass from a rapidity pole gives the displayed increasing breather masses. The kink mass is $M=mn/\pi$, and $k=n$ is an excluded threshold state. No breathers occur at $n=1$; scattering involving a physical second breather requires $n\ge 3$.

For the transmitting product with factors $\cosh [(\theta -i\pi j/N)/2]/\cosh [(\theta +i\pi j/N)/2]$ and prefactor $(-1{)}^N$, the continuous phase on $\theta \ge 0$ is $\pi N-2{\sum }_{j=1}^{N-1}\arctan [\tanh (\theta /2)\tan (\pi j/(2N))]$. Its derivative is $-{\sum }_j\sin (\pi j/N)/[\cosh \theta +\cos (\pi j/N)]$. A Riemann sum at fixed positive rapidity gives $(2N/\pi )\log \tanh (\theta /2)$ at leading order. The endpoints depend on the stated unwrapped branch; setting the high-energy value to zero silently changes the constant.

The second breather is a bound pair of first breathers with constituent shifts $\pm ia$. The bound-state fusion of factorized S-matrices gives $S_{2,1}=S_{1,1}(\vartheta +ia)S_{1,1}(\vartheta -ia)$, which factors as displayed. The nearest physical-strip pole at $\vartheta =ia$ is a crossed-channel exchange of the first breather. At $n=3$ the more distant center pole is double, without changing that nearest-pole interpretation.

At the closest physical-strip pole of $S_{2,1}$, the difference of the analytically continued external momenta has squared mass $m_1^2$. Using $m_2=2m_1\cos a$ makes the identity immediate. Thus the exchanged t-channel one-particle state is the lightest Sine-Gordon breather, not a new member of the spectrum.

For the angular field $u=\beta \varphi$ and coordinates $\xi =m(x+t)/2$, $\eta =m(x-t)/2$, let $w=(v+u)/2$ and $d=(v-u)/2$. The displayed relations imply $u_{\xi \eta }=\sin u$ and $v_{\xi \eta }=\sin v$. They define a Bäcklund transformation with the reciprocal parameter convention used in the Bianchi permutability for sine-Gordon Bäcklund transformations. Physical-field trigonometric arguments include the coupling $\beta$.

Two compatible Bäcklund steps commute after integration constants are matched. The displayed superposition relation constructs their common output algebraically from the seed and the two one-step outputs, in the reciprocal-parameter convention of the Sine-Gordon Bäcklund transformation. It uses the angular field $u=\beta \varphi$. Smooth inverse-tangent branch continuation is required to retain the correct vacuum labels.

Write the transformed angular field as $v=u+2d$. The Sine-Gordon Bäcklund transformation gives $\sin d=a(u_{\xi }+d_{\xi })$, determining $d$ as a formal local derivative expansion in $a$. The displayed exact current identity yields a conservation law at each order. Formal convergence is unnecessary because each coefficient obeys an exact identity on solutions.

The Bäcklund expansion produces local differential-polynomial currents. In the coordinates $\xi =m(x+t)/2$, $\eta =m(x-t)/2$, their charges are $\int (P_j+Q_j)dx$ with vanishing boundary flux. Derivative improvements contribute no new charge. The first nontrivial higher current can be written $P=u_{\xi \xi }^2-u_{\xi }^4/4$, $Q=\cos u{u}_{\xi }^2$. Continuing, and exchanging light-cone directions, produces the infinite higher-spin hierarchy characterizing classical integrability.

With canonical field $\phi =\varphi /\beta$, the quartic interaction has coupling $-m^2{\beta }^2$. Cancelling the vacuum tadpole diagram requires the divergent mass counterterm $\delta m^2=(m^2{\beta }^2/4){\int }_{-\Lambda }^{\Lambda }dk/[2\pi \sqrt{k^2+1}]$. Its kink energy is $\Delta M_{\mathrm{ct}}=4\delta m^2/(m{\beta }^2)$ because $\int (1-\cos {\varphi }_K)dx=4$. This cancels the logarithmic ultraviolet divergence in the one-loop soliton mass correction. Finite parts depend on the renormalization condition.

For real parameters with ${\kappa }_i^2-b_i^2=1$, set $E_i=e^{{\kappa }_{i}x-b_{i}t+{\gamma }_i}$ and $a_{ij}=[({\kappa }_i-{\kappa }_j{)}^2-(b_i-b_j{)}^2]/[({\kappa }_i+{\kappa }_j{)}^2-(b_i+b_j{)}^2]$. Sum ${\prod }_i{E}_i^{{\mu }_i}{\prod }_{i<j}{a}_{ij}^{{\mu }_i{\mu }_j}$ over binary vectors of even parity for $f$ and odd parity for $g$. The Sine-Gordon equation solution is the continuous field $4\arg (f+ig)$. In the all-kink sector, ${\kappa }_i=\cosh {\theta }_i$, $b_i=\sinh {\theta }_i$ and $a_{ij}=-{\tanh }^2[({\theta }_i-{\theta }_j)/2]$. Treating these coefficients as positive would change the solution. Distinct rapidities give separated incoming and outgoing solitons.

The static kink ${\varphi }_K(x)=4\arctan e^{x-a}$ joins adjacent scalar-field vacua $0$ and $2\pi$. It obeys ${\varphi }_K^{\prime }=2\mathrm{sech}(x-a)$ and has topological charge $Q=[\varphi (+\infty )-\varphi (-\infty )]/(2\pi )=1$. In the Sine-Gordon theory normalization with physical mass scale $m$ and coupling $\beta$, its classical mass is $8m/{\beta }^2$. A Lorentz boost gives ${\varphi }_K=4\arctan \exp [(x-vt-a)/\sqrt{1-v^2}]$. Spatial reflection gives an antikink.

This localized two-soliton field has zero net winding and describes an elastic kink-antikink collision for $0<v<1$. Two reciprocal positive Bäcklund parameters produce it from the vacuum. Analytic continuation of $v$ to an imaginary value gives a real Sine-Gordon breather. The overall field sign and spacetime translations change conventions, not the field equation.

The zero-relative-speed limit of the Sine-Gordon kink-antikink scattering solution is a separatrix at twice the rest-kink energy. A Sine-Gordon Bäcklund transformation of the static kink with equal parameter gives $p_x=-p\tanh x$, $p_t=\mathrm{sech}x$ for $p=\tan (\varphi /4)$, hence the displayed solution. The two asymptotic transitions separate logarithmically in time and their speeds tend to zero; this is not a finite-period Sine-Gordon breather.

For $0<v<1$ and $\Gamma =(1-v^2{)}^{-1/2}$, this real two-soliton solution has angular winding $4\pi$. Its separated kink velocities are $\pm v$. At large times the centers satisfy $|m\Gamma x|=|m\Gamma vt|-\log v+o(1)$, giving a right-moving shift $-2\log v/(m\Gamma )$. It follows from Bianchi permutability for sine-Gordon Bäcklund transformations with oppositely signed seed parameters.

For two equal-charge Sine-Gordon kinks with speeds $\pm v$, $0<v<1$, the positive spatial transition satisfies $X_{+}(T)=v|T|-\log v/(m\gamma )+o(1)$, where $\gamma =(1-v^2{)}^{-1/2}$. Labeling a soliton by its preserved rapidity, the right-moving incoming and outgoing intercepts differ by $-2\log v/(m\gamma )$. The soliton time delay at a fixed distant location is minus this shift divided by $v$, giving the displayed negative value. Reversing the entire field gives two antikinks with the same shifts. The sign means an advance relative to extrapolation of the incoming line; labels tied to left and right positions instead exchange velocities during reflection.

The second variation of the Sine-Gordon theory action about its static kink gives ${\Delta }_x=-{\partial }_x^2+1-2{\mathrm{sech}}^{2}x$. Let $A={\partial }_x+\tanh x$. Then ${\Delta }_x=A^{†}A$ and $A{A}^{†}=-{\partial }_x^2+1$. Thus ${\Delta }_x$ is nonnegative, with its translational zero mode of a sine-Gordon kink and a continuum at ${\omega }^2=k^2+1$. The construction is the supersymmetric factorization of the one-soliton potential.

The normalized eigenfunction ${\psi }_0(x)=\mathrm{sech}x/\sqrt{2}$ has zero eigenvalue under the Sine-Gordon kink fluctuation operator. It is proportional to ${\partial }_x{\varphi }_K$ and comes from shifting the collective coordinate of the kink. Its frequency is zero, so it contributes no oscillator zero-point energy; it must be handled separately from a Gaussian functional determinant.

A scalar field configuration eigenstate assigns eigenvalues of the equal-time field operator at every spatial point. These generalized states play the role of position eigenstates for a system of infinitely many coordinates. Inserting their regulated completeness relations on time slices constructs the scalar field path integral.

For a free massive scalar in a Euclidean path integral, inversion of the quadratic kernel gives ${\Delta }_E(p)=1/(p^2+m^2)$. The Minkowski version follows by Wick rotation.

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.

The Gaussian contraction used while integrating out a momentum shell. Its Fourier support lies entirely in that shell. In particular, convolution with a purely low-momentum field is zero. This distinguishes shell Feynman diagrams from unrestricted loop diagrams.

A nonminimal scalar coupling includes an interaction between the field and spacetime curvature, such as $-\xi R{\Phi }^2$ in the Lagrangian density. The Euler-Lagrange field equation for $\mathcal{L}=-\frac{1}{2}(\nabla \Phi {)}^2-\xi R{\Phi }^2$ is $\square \Phi -2\xi R\Phi =0$. The coefficient is convention dependent: an action written with $-\frac{1}{2}\xi R{\Phi }^2$ has $\xi$ instead of $2\xi$.

For two free real scalar fields, the internal rotation $\delta {\varphi }_1=\theta {\varphi }_2$, $\delta {\varphi }_2=-\theta {\varphi }_1$ is a symmetry precisely when their squared masses agree. Its Noether current is $j^{\mu }={\varphi }_2{\partial }^{\mu }{\varphi }_1-{\varphi }_1{\partial }^{\mu }{\varphi }_2$. Without mass degeneracy, ${\partial }_{\mu }{j}^{\mu }=(m_2^2-m_1^2){\varphi }_1{\varphi }_2$.

The equal-time canonical commutation relation is $[\varphi (t,\mathbf{x}),\pi (t,\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y})$ in units $\hslash =1$, with the two field-field and momentum-momentum commutators zero. The quadratic Hamiltonian operator gives $\dot{\varphi }=\pi$ and $\dot{\pi }={\nabla }^2\varphi -m^2\varphi$, hence the Klein-Gordon equation as an operator identity.

For the real scalar field expansion $\varphi =\int d^{3}p[a_{\mathbf{p}}{e}^{-ipx}+a_{\mathbf{p}}^{†}{e}^{ipx}]/((2\pi {)}^3\sqrt{2E_{\mathbf{p}}})$, the equal-time fields extract $a_{\mathbf{p}}=e^{iEt}\int d^{3}x{e}^{-i\mathbf{p}\cdot \mathbf{x}}[\sqrt{E/2}\varphi +i\pi /\sqrt{2E}]$. The two mixed canonical commutation relation terms yield $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, while the other oscillator commutators vanish. This verifies the normalization of the mode expansion directly from the canonical fields.

Phi-fourth theory has a scalar interaction $-\lambda {\varphi }^4/4!$. In four dimensions its coupling is classically marginal, and its one-loop four-point function receives bubble corrections in the three Mandelstam channels.

The interaction vertex $-i\lambda$ and scalar propagator $-i/(p^2-iϵ)$ give one four-point bubble diagram for each of three momentum channels, each with Feynman-diagram symmetry factor $1/2$. The two-point tadpole diagram is zero as a scaleless integral in dimensional regularization. If full proper vertices include the free quadratic kernel, ${\hat{\tau }}_2^{(0)}=-p^2$; the interaction self-energy convention instead starts at zero.

With interaction $\lambda {\varphi }^4/4!$ for one real scalar, the three one-loop four-point bubble channels give the four-dimensional renormalization-group beta function $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$. Its positive sign for $\lambda >0$ excludes asymptotic freedom and produces a perturbative Landau pole on extrapolation.

Phi cubed theory has interaction $-g{\varphi }^3/3!$, giving a cubic vertex $-ig$. In four spacetime dimensions the scalar field has mass dimension one and $g$ has mass dimension one.

The two-vertex cubic bubble in $d=6-ϵ$ dimensions has the displayed amputated pole. A Feynman parameter gives mass $m^2+x(1-x)k^2$, and $\Gamma (-1+ϵ/2)$ supplies the pole. The full propagator receives $D_0^{2}I$, while its inverse receives $-I$. This sign distinguishes a two-point insertion from an inverse-propagator correction.

With additive counterterms $\frac{1}{2}\delta Z(\partial \varphi {)}^2+\frac{1}{2}\delta m^2{\varphi }^2$, their Euclidean propagator insertion is $-(\delta Z{k}^2+\delta m^2)$. These minimal subtraction scheme coefficients cancel the cubic bubble insertion. A bare mass shift also subtracts $m^2\delta Z$ after wavefunction renormalization; it is not the same coefficient as the additive mass counterterm.

A canonically normalized real scalar field in six dimensions has mass dimension two. The cubic coupling is classically a marginal coupling, whereas a local quartic coupling has mass dimension minus two. This theory is perturbatively renormalizable by power counting in quantum field theory, but a real cubic Euclidean potential is unbounded below. Its expansion about a massive Gaussian theory is therefore formal perturbation theory, not a convergent positive functional integral at nonzero real $g$.

For a cubic graph with $E$ external legs, $3V=2I+E$ and $L=I-V+1$. Its superficial degree of divergence is therefore $6L-2I=6-2E$. Divergences in vacuum, one-, two-, and three-point functions can be absorbed into vacuum energy, a linear term, mass, wave-function renormalization, and cubic-coupling counterterms. Higher-point proper diagrams are superficially convergent after subtraction of divergent subgraphs. This establishes perturbative renormalizability, without asserting a nonperturbative positive measure for the unstable real cubic potential.

In a six-dimensional cubic scalar field theory, three labelled box Feynman diagrams give the one-loop local quartic vertex at zero external momentum. With an effective-action term $g_4{\varphi }^4/4!$, their contribution is $\delta g_4=-3g^4{I}_4$. To check its sign and multiplicity, expand the one-loop scalar effective action as $\frac{1}{2}\mathrm{Tr}\log (D+g\varphi )$, where $D=-{\partial }^2+m^2$. Its fourth-order term is $-g^4\mathrm{Tr}(D^{-1}\varphi {)}^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.

For $m>0$ and $0<\Lambda <{\Lambda }_0$, defineThe radial formula uses the area ${\pi }^3$ of the unit five-sphere. Its mass dimension is $6-2r$. As ${\Lambda }_0\to \infty$, $I_1$ diverges quartically, $I_2$ quadratically, $I_3$ logarithmically, and $I_r$ is ultraviolet finite for $r\ge 4$ at fixed $\Lambda ,m$. This follows directly by comparison with $q^{5-2r}$.

The four-point tree-level Feynman diagram has two cubic vertices joined by one propagator. The three ways to divide four labeled external legs into vertex pairs give the $s$, $t$, and $u$ channels. For interaction $+\lambda {\varphi }^3/3!$, the amplitude is $\mathcal{M}=-{\lambda }^2[(s-m^2{)}^{-1}+(t-m^2{)}^{-1}+(u-m^2{)}^{-1}]$, with the Feynman i-epsilon prescription understood. All channels interfere in its modulus squared.

A connected five-point tree in phi cubed theory has three cubic vertices and two internal propagators. For labeled external legs there are fifteen diagrams: choose the leg attached to the middle vertex and partition the other four legs into two unordered pairs.

Phi-six theory has scalar interaction $-\lambda {\varphi }^6/6!$. In four spacetime dimensions its coupling has mass dimension $-2$, and a six-valent vertex contributes the momentum-space factor $-i\lambda$.

A complex scalar field has distinct particle and antiparticle excitations and a global phase symmetry.

A nonlinear complex scalar field theory with a global phase symmetry and a curved target-space kinetic coefficient. In physical coordinates its classical density is displayed above; dimensionless coordinates measured in $M^{-1}$ put a common $M^2$ outside the reduced density. Its regular coordinate domain is ${\lambda }^2|\psi {|}^2<1$. Rotating localized solutions are charged complex sine-Gordon solitons. The local classical density does not by itself specify the treatment of the singular coordinate boundary or a global quantum completion.

The rest-frame field $\psi =\cos \alpha e^{i\sin \alpha t}/[\lambda \cosh (\cos \alpha x)]$ has mass $4M\cos \alpha /{\lambda }^2$ and charge $4[\mathrm{sgn}(\alpha )\pi /2-\alpha ]/{\lambda }^2$, for the generator $\delta \psi =i\psi$ and $\alpha \ne 0$. The conserved phase angle is a collective coordinate with momentum $Q$. Quantization of a periodic soliton coordinate gives integer $Q$ in units $\hslash =1$ and the displayed leading mass formula. On the regular branch $0<|Q|<2\pi /{\lambda }^2$, its concave increasing sine law prevents fragmentation into smaller like-charge states.

At zero rotation the field reaches ${\lambda }^2|\psi {|}^2=1$ at its center. Its energy remains finite, but its charge expression is singular and has the two displayed one-sided limits. The regular local branch excludes this endpoint. A global completion must say whether and how these two limiting charge labels represent a physical state.

An additional integer-level completion identifies the charge labels modulo $k$ and has $k-1$ nonzero sectors. For even $k$, the two maximal labels describe one self-conjugate sector. This interpretation is specified in Dorey and Hollowood, section 2; it is additional to the local classical density.

A free complex scalar field has independent particle and antiparticle mode operators. With $d{\Pi }_p=d^3\mathbf{p}/[(2\pi {)}^{3}2E_p]$, the expansion $\varphi =\int d{\Pi }_p(a{e}^{-ipx}+b^{†}{e}^{ipx})$ and its adjoint reproduce equal-time canonical commutation relations when $[a(\mathbf{p}),a^{†}(\mathbf{q})]=[b(\mathbf{p}),b^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$ and cross commutators vanish. The canonical momenta are $\pi ={\dot{\varphi }}^{†}$, ${\pi }^{†}=\dot{\varphi }$. An overall phase on $b$ can reverse the sign of the antiparticle term without changing the theory.

For the global phase transformation $\delta \varphi =-i\alpha \varphi$, the Noether current is $j^{\mu }=i({\varphi }^{†}{\partial }^{\mu }\varphi -{\partial }^{\mu }{\varphi }^{†}\varphi )$. The Klein-Gordon equation gives ${\partial }_{\mu }{j}^{\mu }=0$. With zero vacuum charge, its normal-ordered charge is $Q=\int d{\Pi }_p(a^{†}a-b^{†}b)$. Thus $[Q,a^{†}]=a^{†}$, $[Q,b^{†}]=-b^{†}$ and $[Q,\varphi ]=-\varphi$. Particle and antiparticle creation carry opposite charges while both increase energy.

Spatial integration of the free Hamiltonian density cancels particle-antiparticle pair terms through $E_p^2-{\mathbf{p}}^2-m^2=0$. The raw Hamiltonian is $\int d{\Pi }_p{E}_p(a^{†}a+b{b}^{†})$. Normal ordering removes the field-independent vacuum energy and gives $H=\int d{\Pi }_p{E}_p(a^{†}a+b^{†}b)$. Both creation operators increase the energy by $E_p$, so a negative-frequency field mode represents a positive-energy antiparticle, not a negative-energy state.

For $\mathcal{L}={\partial }_{\mu }{\psi }^{\ast }{\partial }^{\mu }\psi -V(|\psi {|}^2)$, constant $\alpha$ preserves both terms. This is an internal symmetry of a classical field theory with circle group $U(1)$. A spacetime-dependent phase would introduce derivative terms, so the ordinary-derivative density has only the global symmetry. The associated current is the Noether charge of a complex scalar field construction; the orientation $e^{-i\alpha }$ fixes its overall sign.

For the global phase symmetry of a complex scalar field oriented as $\delta \psi =-i\epsilon \psi$, the Noether current is $j^{\mu }=i({\psi }^{\ast }{\partial }^{\mu }\psi -\psi {\partial }^{\mu }{\psi }^{\ast })$. The Euler-Lagrange field equations $\square \psi +V^{\prime }(|\psi {|}^2)\psi =0$ and its complex conjugate give ${\partial }_{\mu }{j}^{\mu }=i({\psi }^{\ast }\square \psi -\psi \square {\psi }^{\ast })=0$ for a real differentiable potential. Integrating $j^0$ gives $Q$ when boundary flux vanishes. A charged scalar coupled to electromagnetism has electric charge $qQ$ with the chosen charge normalization; without that physical identification it is an internal charge.