Scalar field theory

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''For the pseudoscientific "scalar field theory", see "Scalar field theory (pseudoscience)"

In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. Such a field is distinguished by its invariance under a Lorentz transformation, hence the name "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 certainly do appear in the effective field theory descriptions of many physical phenomena. 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.

Although scalar fields are Lorentz scalars, they may transform nontrivially under other symmetries, such as flavour or isospin. For example, the pion is invariant under the restricted Lorentz group, but is an isospin triplet furthermore (meaning it transforms like a three component vector under the SU(2) isospin symmetry). Furthermore, it picks up a negative phase under parity inversion, so it transforms nontrivially under the full Lorentz group; such particles are called pseudoscalar rather than scalar. Most mesons are pseudoscalar particles.

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, $eta\_\{munu\}$, 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 quantized version is known as a free theory, and sometimes this nomenclature is used even in the classical case.)

Action and equation of motion

The action for the linear, relativistic scalar field theory is

$mathcal\{S\}=int\; mathrm\{d\}^\{D-1\}x\; mathrm\{d\}t\; mathcal\{L\}\; =\; int\; mathrm\{d\}^\{D-1\}x\; mathrm\{d\}t$

[frac{1}{2}eta^{munu}partial_muphipartial_nuphi -frac{1}{2} m^2phi^2]=int mathrm{d}^{D-1}x mathrm{d}t [frac{1}{2}(partial_tphi)^2-frac{1}{2}delta^{ij}partial_iphipartial_jphi -frac{1}{2} m^2phi^2], where $mathcal\{L\}$ is known as a Lagrangian density. This is an example of a quadractic 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 $phi$:

$eta^\{munu\}partial\_mupartial\_nuphi+m^2phi=partial^2\_tphi-nabla^2phi$

+m^2phi=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 usual generalization of the linear theory above is to add terms polynomial in $phi$ to the equations of motion. Such a theory is sometimes said to be interacting, again influenced by quantum field theoretical language.

Action and equation of motion

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^{munu}partial_muphipartial_nuphi -V(phi) right]

$=int\; mathrm\{d\}^\{D-1\}x\; ,\; mathrm\{d\}t\; left[frac\{1\}\{2\}(partial\_tphi)^2-\; frac\{1\}\{2\}delta^\{ij\}partial\_iphipartial\_jphi\; -$

frac{1}{2}m^2phi^2-sum_{n=3}^infty g_nphi^n right] where $V(phi)$ is known as a potential. The corresponding equation of motion is

$eta^\{munu\}partial\_mupartial\_nuphi+V\text{'}(phi)=partial^2\_tphi-nabla^2phi$

+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.

Scaling dimension of Φ

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

$xrightarrowlambda\; x$

$phirightarrowlambda^\{-Delta\}phi$

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

$xrightarrowlambda\; x$

$phirightarrowlambda^\{-Delta\}phi$

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.

Conformal invariance

A transformation

$xrightarrow\; tilde\{x\}(x)$

is said to be conformal if the transformation satisfies

$frac\{partialtilde\{x^mu\}\}\{partial\; x^rho\}frac\{partialtilde\{x^nu\}\}\{partial$

x^sigma}eta_{munu}=lambda^2(x)eta_{rhosigma} for some function $lambda^2(x)$. The conformal group contains as subgroups the isometries of the metric $eta\_\{munu\}$ (the Poincare 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

Massive $phi^4$ theory illustrates a number of interesting phenomena in scalar field theory. The Lagrangian density is

$mathcal\{L\}=frac\{1\}\{2\}(partial\_tphi)^2-frac\{1\}\{2\}delta^\{ij\}partial\_iphipartial\_jphi\; -$

frac{1}{2}m^2phi^2-gphi^4.

Spontaneous symmetry breaking

This Lagrangian has a $Z\_2$ symmetry, where the transformation is :$phirightarrow-phi$ This is an example of an internal symmetry, in contrast to a space-time symmetry.

If $m^2$ is positive, the potential $V(phi)=frac\{1\}\{2\}m^2phi^2\; +gphi^4$ 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 $V(phi)=m^2phi^2\; +gphi^4$ 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

$phi(vec\{x\},t)=pmfrac\{m\}\{2sqrt\{g\}\}tanhleft(frac\{m(x-x\_0)\}\{sqrt\{2\}\}right)$

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.

Action

The most general action one usually considers for a complex scalar field takes the following form:

$mathcal\{S\}=int\; mathrm\{d\}^\{D-1\}x\; ,\; mathrm\{d\}t$

mathcal{L} = int mathrm{d}^{D-1}x , mathrm{d}t left[frac{1}{2}eta^{munu}partial_muphi^*partial_nuphi -V(|phi|^2)right] It has a U(1) symmetry, whose action on the space of fields rotates $phirightarrow\; e^\{ialpha\}phi$, for some real angle $alpha$.

Mexican hat

The mexican hat potential is analogous to the double-well potential in real scalar field theory. It occurs when the potential $V(|phi|^2)$ is of the form :$V(|phi|^2)=\; m^2|phi|^2\; +g\; |phi|^4$ where $m^2$ is negative. It is so-called because it resembles a sombrero.

A theory with such a potential also exhibits spontaneous symmetry breaking, because the vacua are not invariant under the action of the internal symmetry $phirightarrow\; e^\{ialpha\}phi$. Because this symmetry is continuous, one can apply Goldstone's theorem.

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

$mathcal\{L\}=frac\{1\}\{2\}eta^\{munu\}partial\_muphicdotpartial\_nuphi\; -V(phicdotphi)$

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

Quantum scalar field theory

Quantum field theory is usually thought of as the quantization of a classical field theory. When the canonical quantization procedure is applied to a classical field theory, the classical field variable, $phi(x)$ becomes an operator, acting on states in the Hilbert space of the quantum field theory.

Note that the classical field equations bear some resemblance to quantum mechanical relativistic wave equations. An example given above is the Klein-Gordon equation, which was originally interpreted as a wave equation. For this reason, the process of canonical quantization of a field theory was called second quantization in the early literature.

Partition function

In the path integral formulation of quantum field theory, one computes objects of interest in quantum field theory by means of the path integral, also known as the partition function. The partition function of a real quantum scalar field theory is defined as a functional integration over possible field configurations:

$Z[J]\; =\; int\; D[phi]\; expleft(frac\{-i\}\{hbar\}\; int\; d^\{D-1\}xdtleft[mathcal\{L\}(phi(x))$

+ J(x)phi(x)right] right)

where J is an external current, and $mathcal\{L\}(phi(x))$ could be any one of the classical Lagrangians described above. The choice of classical Lagrangian defines the field theory.

Note that computing a partition function is a far from trivial process for most choices of Lagrangian, and usually one has to make some kind of approximation. An exception is the case of the free theory, where the Lagrangian is quadratic in the fields, $phi(x)$. When one adds higher-order terms in $phi$ to the Lagrangian, the most usual approximation is perturbation theory, where one assumes that the coefficients of the higher-order terms are small. These coefficients are labelled $g\_n$ in the description of the classical non-linear theory above, and are known as coupling constants in the corresponding quantum field theory. In some sense, coupling constants describe the strength of particle interactions in a theory.

Expectation values

The time ordered vacuum expectation value of an operator $F(phi)$ is given by

$leftlangle\; Frightrangle=frac\{int\; mathcal\{D\}phi\; F[phi]e^\{iS[phi]\}\}\{intmathcal\{D\}phi\; e^\{iS[phi]\}\}.$

Correlation functions

Computing correlation functions in a quantum field theory is part of the procedure to compute S-matrix elements, which yield predictions for particle scattering. The time ordered correlation functions of the fields, $phi(x)$, can be expressed in terms of functional derivatives of the path integral:

$\{leftlangle\; phi(x\_1)cdots\; phi(x\_n)rightrangle\}\_J=(-i)^n\; ,\; frac\{1\}\{Z[J]\}\; ,\; frac\{delta^n\; Z[J]\}\{delta\; J(x\_1)\; cdots\; delta\; J(x\_n)\}$.

If required, the external current can be set to zero after taking the functional derivatives.

Renormalization

Beta-function

A beta-function, β(g), encodes the dependence of a coupling parameter, g, on the energy scale, μ of a given physical process. It is defined by the relation:

$beta(g)\; =\; mu,frac\{partial\; g\}\{partial\; mu\}$.

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 some kind of approximation scheme. An example is perturbation theory, where one assumes that the coupling parameters are 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).

Example: $phi^4$ theory

The Lagrangian density for $phi^4$ quantum scalar field theory is:

$mathcal\{L\}=frac\{1\}\{2\}eta^\{munu\}partial\_muphipartial\_nuphi\; -$

frac{1}{2}m^2phi^2-frac{g}{4!}phi^4. Note that there is a factor of $frac\{1\}\{4!\}$ in the quartic (interaction) term, relative to the same term in the classical Lagrangian. This difference is purely conventional, and derives from typical factors of 4! that appear when computing Feynman diagrams in perturbation theory.

The beta-function at one loop for the coupling parameter, g, is:

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

We note that the sign in front of the lowest order term is positive. One cannot rely on this lowest order result at large couplings, but if one were to naively follow perturbation theory then this sign would indicate the presence of a Landau pole at finite energy.

Renormalizability

D=4

D=3

D=2

Scale invariance

Fixed points of the RG flow

epsilon expansion

References

Almost any textbook on quantum field theory (QFT) will contain a description of both classical and quantum scalar field theory. The following are reliable sources:

• Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory, Westview Press (1995). A standard introductory text, covering many topics in QFT.
• Weinberg, Steven ; The Quantum Theory of Fields, (3 volumes) Cambridge University Press (1995). A monumental treatise on QFT.
• Srednicki, Mark; Quantum Field Theory, Cambridge University Press (2007). Very clearly and fully explained introduction to the topic. Web page here
• Zinn-Justin, Jean ; Quantum Field Theory and Critical Phenomena, Oxford University Press (2002). Emphasis on the renormalization group and extensive discussion of the epsilon expansion and related topics.

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