Scalar field theory

In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field. The quanta of the quantized scalar field are spin-zero particles, and as such are bosons.

No fundamental scalar fields have been observed in nature, though the Higgs boson may yet prove the first example. However, scalar fields appear in the effective field theory descriptions of many physical phenomena. An example is the pion, which is actually a "pseudoscalar", which means it is not invariant under parity transformations which invert the spatial directions, distinguishing it from a true scalar, which is parity-invariant. Because of the relative simplicity of the mathematics involved, scalar fields are often the first field introduced to a student of classical or quantum field theory.

In this article, the repeated index notation indicates the Einstein summation convention for summation over repeated indices. The theories described are defined in flat, D-dimensional Minkowski space, with (D-1) spatial dimension and one time dimension and are, by construction, relativistically covariant. The Minkowski space metric, $\eta _{\mu \nu }$, has a particularly simple form: it is diagonal, and here we use the + − − − sign convention.

Classical scalar field theory

Linear (free) theory

The most basic scalar field theory is the linear theory. The action for the free relativistic scalar field theory is

${\mathcal {S}}=\int \mathrm {d} ^{D-1}x\mathrm {d} t{\mathcal {L}}=\int \mathrm {d} ^{D-1}x\mathrm {d} t\left[{\frac {1}{2}}\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right]$

$=\int \mathrm {d} ^{D-1}x\mathrm {d} t\left[{\frac {1}{2}}(\partial _{t}\phi )^{2}-{\frac {1}{2}}\delta ^{ij}\partial _{i}\phi \partial _{j}\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right],$

where ${\mathcal {L}}$ is known as a Lagrangian density. This is an example of a quadratic action, since each of the terms is quadratic in the field, $\phi$. The term proportional to $m^{2}$ is sometimes known as a mass term, due to its interpretation in the quantized version of this theory in terms of particle mass.

The equation of motion for this theory is obtained by extremizing the action above. It takes the following form, linear in $\phi$:

$\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\phi +m^{2}\phi =\partial _{t}^{2}\phi -\nabla ^{2}\phi +m^{2}\phi =0$

Note that this is the same as the Klein–Gordon equation, but that here the interpretation is as a classical field equation, rather than as a quantum mechanical wave equation.

Nonlinear (interacting) theory

The most common generalization of the linear theory above is to add a scalar potential $V(\phi )$ to the equations of motion, where typically, V is a polynomial in φ of order 3 or more (often a monomial). Such a theory is sometimes said to be interacting, because the Euler-Lagrange equation is now nonlinear, implying a self-interaction. The action for the most general such theory is

${\mathcal {S}}=\int \mathrm {d} ^{D-1}x\,\mathrm {d} t{\mathcal {L}}=\int \mathrm {d} ^{D-1}x\mathrm {d} t\left[{\frac {1}{2}}\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi -V(\phi )\right]$

$=\int \mathrm {d} ^{D-1}x\,\mathrm {d} t\left[{\frac {1}{2}}(\partial _{t}\phi )^{2}-{\frac {1}{2}}\delta ^{ij}\partial _{i}\phi \partial _{j}\phi -{\frac {1}{2}}m^{2}\phi ^{2}-\sum _{n=3}^{\infty }{\frac {1}{n!}}g_{n}\phi ^{n}\right]$

The n! factors in the expansion are introduced because they are useful in the Feynman diagram expansion of the quantum theory, as described below. The corresponding Euler-Lagrange equation of motion is

$\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\phi +V'(\phi )=\partial _{t}^{2}\phi -\nabla ^{2}\phi +V'(\phi )=0$.

Dimensional analysis and scaling

Physical quantities in these scalar field theories may have dimensions of length, time or mass, or some combination of the three. However, in a relativistic theory, any quantity t, with dimensions of time, can be 'converted' into a length, $l=ct$, by using the velocity of light, c.

Similarly, any length l is equivalent to an inverse mass, $l={\frac {\hbar }{mc}}$, using Planck's constant, $\hbar$. Heuristically, one can think of a time as a length, or either time or length as an inverse mass. In short, one can think of the dimensions of any physical quantity as defined in terms of just one independent dimension, rather than in terms of all three. This is most often termed the mass dimension of the quantity.

One objection is that this theory is classical, and therefore it is not obvious that Planck's constant should be a part of the theory at all. In a sense this is a valid objection, and if desired one can indeed recast the theory without mass dimensions at all. However, this would be at the expense of making the connection with the quantum scalar field slightly more obscure. Given that one has dimensions of mass, Planck's constant is thought of here as an essentially arbitrary fixed quantity with dimensions appropriate to convert between mass and inverse length. This is consistent with the Feynman path integral approach to quantization, where the only reason for Planck's constant to appear stems from the same type of dimensional argument, since the action must be divided by some parameter with these dimensions to render the phase dimensionless.

Scaling Dimension

The classical scaling dimension, or mass dimension, $\Delta$, of $\phi$ describes the transformation of the field under a rescaling of coordinates:

$x\rightarrow \lambda x$

$\phi \rightarrow \lambda ^{-\Delta }\phi$

The units of action are the same as the units of $\hbar$, and so the action itself has zero mass dimension. This fixes the scaling dimension of $\phi$ to be

$\Delta ={\frac {D-2}{2}}$.

Scale invariance

There is a specific sense in which some scalar field theories are scale-invariant. While the actions above are all constructed to have zero mass dimension, not all actions are invariant under the scaling transformation

$x\rightarrow \lambda x$

$\phi \rightarrow \lambda ^{-\Delta }\phi$

The reason that not all actions are invariant is that one usually thinks of the parameters m and $g_{n}$ as fixed quantities, which are not rescaled under the transformation above. The condition for a scalar field theory to be scale invariant is then quite obvious: all of the parameters appearing in the action should be dimensionless quantities. In other words, a scale invariant theory is one without any fixed length scale (or equivalently, mass scale) in the theory.

For a scalar field theory with D spacetime dimensions, the only dimensionless parameter $g_{n}$ satisfies $n={\frac {2D}{D-2}}$. For example, in D=4 only $g_{4}$ is classically dimensionless, and so the only classically scale-invariant scalar field theory in $D=4$ is the massless $\phi ^{4}$ theory. Classical scale invariance normally does not imply quantum scale invariance. See the discussion of the beta function below.

Conformal invariance

A transformation

$x\rightarrow {\tilde {x}}(x)$

is said to be conformal if the transformation satisfies

${\frac {\partial {\tilde {x^{\mu }}}}{\partial x^{\rho }}}{\frac {\partial {\tilde {x^{\nu }}}}{\partial x^{\sigma }}}\eta _{\mu \nu }=\lambda ^{2}(x)\eta _{\rho \sigma }$

for some function $\lambda ^{2}(x)$. The conformal group contains as subgroups the isometries of the metric $\eta _{\mu \nu }$ (the Poincaré group) and also the scaling transformations (or dilatations) considered above. In fact, the scale-invariant theories in the previous section are also conformally-invariant.

φ⁴ theory

See also: Quartic interaction

Massive $\phi ^{4}$ theory illustrates a number of interesting phenomena in scalar field theory.

The Lagrangian density is

${\mathcal {L}}={\frac {1}{2}}(\partial _{t}\phi )^{2}-{\frac {1}{2}}\delta ^{ij}\partial _{i}\phi \partial _{j}\phi -{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {g}{4!}}\phi ^{4}.$

Spontaneous symmetry breaking

This Lagrangian has a $Z_{2}$ symmetry under the transformation $\phi \rightarrow -\phi$

This is an example of an internal symmetry, in contrast to a space-time symmetry.

If $m^{2}$ is positive, the potential $V(\phi )={\frac {1}{2}}m^{2}\phi ^{2}+{\frac {g}{4!}}\phi ^{4}$ has a single minimum, at the origin. The solution $\phi =0$ is clearly invariant under the $Z_{2}$ symmetry. Conversely, if $m^{2}$ is negative, then one can readily see that the potential $\,V(\phi )={\frac {1}{2}}m^{2}\phi ^{2}+{\frac {g}{4!}}\phi ^{4}\!$ has two minima. This is known as a double well potential, and the lowest energy states (known as the vacua, in quantum field theoretical language) in such a theory are not invariant under the $Z_{2}$ symmetry of the action (in fact it maps each of the two vacua into the other). In this case, the $Z_{2}$ symmetry is said to be spontaneously broken.

Kink solutions

The $\phi ^{4}$ theory with a negative $m^{2}$ also has a kink solution, which is a canonical example of a soliton. Such a solution is of the form

$\phi ({\vec {x}},t)=\pm {\frac {m}{2{\sqrt {g}}}}\tanh \left({\frac {m(x-x_{0})}{\sqrt {2}}}\right)$

where x is one of the spatial variables ($\phi$ is taken to be independent of t, and the remaining spatial variables). The solution interpolates between the two different vacua of the double well potential. It is not possible to deform the kink into a constant solution without passing through a solution of infinite energy, and for this reason the kink is said to be stable. For $D>2$, i.e. theories with more than one spatial dimension, this solution is called a domain wall.

Another well-known example of a scalar field theory with kink solutions is the sine-Gordon theory.

Complex scalar field theory

In a complex scalar field theory, the scalar field takes values in the complex numbers, rather than the real numbers. The action considered normally takes the form

${\mathcal {S}}=\int \mathrm {d} ^{D-1}x\,\mathrm {d} t{\mathcal {L}}=\int \mathrm {d} ^{D-1}x\,\mathrm {d} t\left[\eta ^{\mu \nu }\partial _{\mu }\phi ^{*}\partial _{\nu }\phi -V(|\phi |^{2})\right]$

This has a U(1) symmetry, whose action on the space of fields rotates $\phi \rightarrow e^{i\alpha }\phi$, for some real phase angle $\alpha$.

As for the real scalar field, spontaneous symmetry breaking is found if m² is negative. This gives rise to a Mexican hat potential which is analogous to the double-well potential in real scalar field theory, but now the choice of vacuum breaks a continuous U(1) symmetry instead of a discrete one. This leads to a Goldstone boson.

O(N) theory

One can express the complex scalar field theory in terms of two real fields, $\phi ^{1}=Re{\phi }$ and $\phi ^{2}=Im{\phi }$ which transform in the vector representation of the $U(1)=O(2)$ internal symmetry. Although such fields transform as a vector under the internal symmetry, they are still Lorentz scalars. This can be generalised to a theory of N scalar fields transforming in the vector representation of the O(N) symmetry. The Lagrangian for an O(N)-invariant scalar field theory is typically of the form

${\mathcal {L}}={\frac {1}{2}}\eta ^{\mu \nu }\partial _{\mu }\phi \cdot \partial _{\nu }\phi -V(\phi \cdot \phi )$

using an appropriate $O(N)$-invariant inner product.

Quantum scalar field theory

In quantum field theory, the fields, and all observables constructed from them, are replaced by quantum operators on a Hilbert space. This Hilbert space is built on a vacuum state, and dynamics are governed by a Hamiltonian, a positive operator which annihilates the vacuum. A construction of a quantum scalar field theory may be found in the canonical quantization article, which uses canonical commutation relations among the fields as a basis for the construction. In brief, the basic variables are the field φ and its canonical momentum π. Both fields are Hermitian. At spatial points ${\vec {x}},{\vec {y}}$ at equal times, the canonical commutation relations are given by

$[\phi ({\vec {x}}),\phi ({\vec {y}})]=[\pi ({\vec {x}}),\pi ({\vec {y}})]=0,\ \ \ \ [\phi ({\vec {x}}),\pi ({\vec {y}})]=i\delta ({\vec {x}}-{\vec {y}}),$

and the free Hamiltonian is

$H=\int d^{3}x\left[{1 \over 2}\pi ^{2}+{1 \over 2}(\nabla \phi )^{2}+{m^{2} \over 2}\phi ^{2}\right].$

A spatial Fourier transform leads to momentum space fields

${\tilde {\phi }}({\vec {k}})=\int d^{3}xe^{-i{\vec {k}}\cdot {\vec {x}}}\phi ({\vec {x}}),\ \ \ \ {\tilde {\pi }}({\vec {k}})=\int d^{3}xe^{-i{\vec {k}}\cdot {\vec {x}}}\pi ({\vec {x}})$

which are used to define annihilation and creation operators

$a({\vec {k}})=\left(E{\tilde {\phi }}({\vec {k}})+i{\tilde {\pi }}({\vec {k}})\right),\ \ \ \ a^{\dagger }({\vec {k}})=\left(E{\tilde {\phi }}({\vec {k}})-i{\tilde {\pi }}({\vec {k}})\right),$

where $E={\sqrt {k^{2}+m^{2}}}$. These operators satisfy the commutation relations

$[a({\vec {k}}_{1}),a({\vec {k}}_{2})]=[a^{\dagger }({\vec {k}}_{1}),a^{\dagger }({\vec {k}}_{2})]=0,\ \ \ \ [a({\vec {k}}_{1}),a^{\dagger }({\vec {k}}_{2})]=(2\pi )^{3}2E\delta ({\vec {k}}_{1}-{\vec {k}}_{2}).$

The state |0> annihilated by all of the operators a is identified as the bare vacuum, and a particle with momentum ${\vec {k}}$ is created by applying $a^{\dagger }({\vec {k}})$ to the vacuum. Applying all possible combinations of creation operators to the vacuum constructs the Hilbert space. This construction is called Fock space. The vacuum is annihilated by the Hamiltonian

$H=\int {d^{3}k \over (2\pi )^{3}}{\frac {1}{2}}a^{\dagger }({\vec {k}})a({\vec {k}}),$

where the zero-point energy has been removed by Wick ordering. (See canonical quantization.)

Interactions can be included by adding an interaction Hamiltonian. For a φ⁴ theory, this corresponds to adding a Wick ordered term g:φ⁴:/4! to the Hamiltonian, and integrating over x. Scattering amplitudes may be calculated from this Hamiltonian in the interaction picture. These are constructed in perturbation theory by means of the Dyson series, which gives the time-ordered products, or n-particle Green's functions $\langle 0|{\mathcal {T}}\{{\phi }(x_{1})\cdots {\phi }(x_{n})\}|0\rangle$ as described in the Dyson series article. The Green's functions may also be obtained from a generating function that is constructed as a solution to the Schwinger-Dyson equation.

Feynman Path Integral

The Feynman diagram expansion may be obtained also from the Feynman path integral formulation. The time ordered vacuum expectation values of polynomials in φ, known as the n-particle Green's functions, are constructed by integrating over all possible fields, normalized by the vacuum expectation value with no external fields,

$\langle 0|{\mathcal {T}}\{{\phi }(x_{1})\cdots {\phi }(x_{n})\}|0\rangle ={\frac {\int {\mathcal {D}}\phi \phi (x_{1})\cdots \phi (x_{n})e^{i\int d^{4}x\left({1 \over 2}\partial ^{\mu }\phi \partial _{\mu }\phi -{m^{2} \over 2}\phi ^{2}-{g \over 4!}\phi ^{4}\right)}}{\int {\mathcal {D}}\phi e^{i\int d^{4}x\left({1 \over 2}\partial ^{\mu }\phi \partial _{\mu }\phi -{m^{2} \over 2}\phi ^{2}-{g \over 4!}\phi ^{4}\right)}}}.$

All of these Green's functions may be obtained by expanding the exponential in J(x)φ(x) in the generating function

$Z[J]=\int {\mathcal {D}}\phi e^{i\int d^{4}x\left({1 \over 2}\partial ^{\mu }\phi \partial _{\mu }\phi -{m^{2} \over 2}\phi ^{2}-{g \over 4!}\phi ^{4}+J\phi \right)}=Z[0]\sum _{n=0}^{\infty }{\frac {i^{n}J(x_{1})\cdots J(x_{n})}{n!}}\langle 0|{\mathcal {T}}\{{\phi }(x_{1})\cdots {\phi }(x_{n})\}|0\rangle .$

A Wick rotation may be applied to make time imaginary. Changing the signature to (++++) then turns the Feynman integral into a statistical mechanics partition function in Euclidean space,

$Z[J]=\int {\mathcal {D}}\phi e^{-\int d^{4}x\left({1 \over 2}(\nabla \phi )^{2}+{m^{2} \over 2}\phi ^{2}+{g \over 4!}\phi ^{4}+J\phi \right)}.$

Normally, this is applied to the scattering of particles with fixed momenta, in which case, a Fourier transform is useful, giving instead

${\tilde {Z}}[{\tilde {J}}]=\int {\mathcal {D}}{\tilde {\phi }}e^{-\int d^{4}p\left({1 \over 2}(p^{2}+m^{2}){\tilde {\phi }}^{2}+{\lambda \over 4!}{\tilde {\phi }}^{4}-{\tilde {J}}{\tilde {\phi }}\right)}.$

The standard trick to evaluate this functional integral is to write it as a product of exponential factors, schematically,

${\tilde {Z}}[{\tilde {J}}]\sim \int {\mathcal {D}}{\tilde {\phi }}\prod _{p}\left[e^{-(p^{2}+m^{2}){\tilde {\phi }}^{2}/2}e^{-g{\tilde {\phi }}^{4}/4!}e^{{\tilde {J}}{\tilde {\phi }}}\right].$

The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically. The integral with λ = 0 can be treated as a product of infinitely many elementary Gaussian integrals, and the result may be expressed as a sum of Feynman diagrams, calculated using the following Feynman rules:

• Each field ${\tilde {\phi }}(p)$ in the n-point Euclidean Green's function is represented by an external line (half-edge) in the graph, and associated with momentum p.
• Each vertex is represented by a factor -g.
• At a given order g^k, all diagrams with n external lines and k vertices are constructed such that the momenta flowing into each vertex is zero. Each internal line is represented by a propagator 1/(q² + m²), where q is the momentum flowing through that line.
• Any unconstrained momenta are integrated over all values.
• The result is divided by a symmetry factor, which is the number of ways the lines and vertices of the graph can be rearranged without changing its connectivity.
• Do not include graphs containing "vacuum bubbles", connected subgraphs with no external lines.

The last rule takes into account the effect of dividing by ${\tilde {Z}}[0]$. The Minkowski-space Feynman rules are similar, except that each vertex is represented by -ig, while each internal line is represented by a propagator i/(q²-m² + iε), where the ε term represents the small Wick rotation needed to make the Minkowski-space Gaussian integral converge.

Renormalization

The integrals over unconstrained momenta, called "loop integrals", in the Feynman graphs typically diverge. This is normally handled by renormalization, which is a procedure of adding divergent counter-terms to the Lagrangian in such a way that the diagrams constructed from the original Lagrangian and counter-terms is finite. A renormalization scale must be introduced in the process, and the coupling constant and mass become dependent upon it.

The dependence of a coupling constant g on the scale λ is encoded by a beta function, β(g), defined by the relation

$\beta (g)=\lambda \,{\frac {\partial g}{\partial \lambda }}$

This dependence on the energy scale is known as the running of the coupling parameter, and theory of this kind of scale-dependence in quantum field theory is described by the renormalization group.

Beta-functions are usually computed in an approximation scheme, most commonly perturbation theory, where one assumes that the coupling constant is small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher loop contributions, due to the number of loops in the corresponding Feynman graphs).

The beta-function at one loop (the first perturbative contribution) for the $\phi ^{4}$ theory is

$\beta (g)={\frac {3}{16\pi ^{2}}}g^{2}+O(g^{3})$

The fact that the sign in front of the lowest-order term is positive suggests that the coupling constant increases with energy. If this behavior persists at large couplings, this would indicate the presence of a Landau pole at finite energy, or quantum triviality. The question can only be answered non-perturbatively, since it involves strong coupling.

A quantum field theory is trivial when the running coupling, computed through its beta function, goes to zero when the cutoff is removed. Consequently, the propagator becomes that of a free particle and the field is no longer interacting. Alternatively, the field theory may be interpreted as an effective theory, in which the cutoff is not removed, giving finite interactions but leading to a Landau pole at some energy scale. For a φ⁴ interaction, Michael Aizenman proved that the theory is indeed trivial for space-time dimension $D\geq 5$. For $D=4$ the triviality has yet to be proven rigorously, but lattice computations have confirmed this. (See Landau pole for details and references.) This fact is relevant as the Higgs field, for which triviality bounds are used to set limits on the Higgs mass, based on the new physics must enter at a higher scale (perhaps the Planck scale) to prevent the Landau pole from being reached.

See also

• Renormalization
• Quantum triviality
• Landau pole

Further reading

• Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory, Westview Press (1995)
• Weinberg, Steven ; The Quantum Theory of Fields, (3 volumes) Cambridge University Press (1995)
• Srednicki, Mark; Quantum Field Theory, Cambridge University Press (2007)
• Zinn-Justin, Jean ; Quantum Field Theory and Critical Phenomena, Oxford University Press (2002)

External links

• Pedagogic Aides to Quantum Field Theory Click on the link for Chap. 3 to find an extensive, simplified introduction to scalars in relativistic quantum mechanics and quantum field theory.
• 't Hooft, G., "The Conceptual Basis of Quantum Field Theory" (online version).