View source for Scalar field theory

In [[theoretical physics]], '''scalar field theory''' can refer to a relativistically invariant [[Classical field theory|classical]] or [[Quantum field theory|quantum theory]] of [[scalar field]]s. A scalar field is invariant under any [[Lorentz transformation]].<ref>i.e., it transforms under the trivial [[Representation theory of the Lorentz group#Common representations|{{math|(0, 0)}}]]-representation of the Lorentz group, leaving the value of the field at any spacetime point unchanged, in contrast to a [[vector field|vector]] or [[tensor field]], or more generally, spinor-tensors, whose components undergo a mix under Lorentz transformations. Since particle or field spin by definition is determined by the Lorentz representation under which it transforms, all scalar (and pseudoscalar) fields and particles have spin zero, and are as such [[boson]]ic by the [[spin statistics theorem]]. See {{harvnb|Weinberg|1995|loc=Chapter 5}}</ref>

The only fundamental scalar quantum field that has been observed in nature is the [[Higgs field]]. However, scalar quantum fields feature in the [[effective field theory]] descriptions of many physical phenomena. An example is the [[pion]], which is actually a [[pseudoscalar]].<ref>This  means it is not invariant under [[parity transformation]]s which invert the spatial directions, distinguishing it from a true scalar, which is parity-invariant.See {{harvnb|Weinberg|1998|loc=Chapter 19}}</ref>

Since they do not involve [[Photon polarization#Polarization states|polarization]] complications, scalar fields are often the easiest to appreciate [[Canonical quantization#Real scalar field|second quantization]] through. For this reason, scalar field theories are  often used for purposes of introduction of novel concepts and techniques.<ref>{{cite book 
 |last=Brown
 |first=Lowell S.
 |authorlink=Lowell S. Brown
 |title=Quantum Field Theory
 |publisher=[[Cambridge University Press]]
 |year=1994
 |isbn=978-0-521-46946-3 
}} Ch 3.</ref>

The [[metric signature|signature of the metric]] employed below is {{math|(+, −, −, −)}}.

==Classical scalar field theory==
{{further|Lagrangian (field theory)}}
A general reference for this section is Ramond, Pierre (2001-12-21). Field Theory: A Modern Primer (Second Edition). USA: Westview Press. {{isbn|0-201-30450-3}},  Ch 1.

===Linear (free) theory===
The most basic scalar field theory is the [[linear]] theory.  Through the Fourier decomposition of the fields, it represents the [[Normal mode#Coupled oscillators|normal modes]] of an [[Quantum_harmonic_oscillator#Harmonic_oscillators_lattice:_phonons|infinity of coupled oscillators]] where the continuum limit of the oscillator index ''i'' is now denoted by {{mvar|x}}. The [[Action (physics)|action]] for the free [[theory of relativity|relativistic]] scalar field theory is then
:<math>\begin{align}
  \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] \\[6pt]
              &= \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],
\end{align}</math>

where <math>\mathcal{L}</math> is known as a [[Lagrangian density]]; {{math|''d''<sup>4&minus;1</sup>''x'' ≡ ''dx'' ⋅ ''dy'' ⋅ ''dz'' ≡ ''dx''<sup>1</sup> ⋅ ''dx''<sup>2</sup> ⋅ ''dx''<sup>3</sup>}} for the three spatial coordinates; {{math|''δ<sup>ij</sup>''}} is the [[Kronecker delta]] function; and {{math|''∂''<sub>''ρ''</sub> {{=}} ''∂''/''∂x''<sup>''ρ''</sup>}}  for the {{mvar|''ρ''}}-th coordinate {{math|''x''<sup>''ρ''</sup>}}.

This is an example of a quadratic action, since each of the terms is quadratic in the field, {{mvar|φ}}. The term proportional to {{math|''m''<sup>2</sup>}} is sometimes known as a mass term, due to its subsequent interpretation, in the quantized version of this theory, in terms of particle mass.

The equation of motion for this theory is obtained by [[Euler–Lagrange|extremizing]] the action above. It takes the following form, linear in  {{mvar|φ}},  

:<math>\eta^{\mu\nu}\partial_\mu\partial_\nu\phi+m^2\phi=\partial^2_t\phi-\nabla^2\phi+m^2\phi=0 ~,</math>

where ∇<sup>2</sup> is the [[Del#Laplacian|Laplace operator]]. This is the [[Klein–Gordon equation]], with the interpretation  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]] {{math|''V''(Φ)}} to the Lagrangian, where typically, in addition to a mass term, ''V'' is a polynomial in {{mvar|Φ}}. Such a theory is sometimes said to be [[interaction|interacting]], because the Euler-Lagrange equation is now nonlinear, implying a [[Self-energy|self-interaction]]. The action for the most general such theory is

:<math>\begin{align}
  \mathcal{S} &= \int \mathrm{d}^{D-1}x \, \mathrm{d}t \mathcal{L} \\[3pt]
              &= \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] \\[3pt]
              &= \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]
\end{align}</math>

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 now
:<math>\eta^{\mu\nu} \partial_\mu \partial_\nu\phi + V'(\phi) = \partial^2_t \phi - \nabla^2 \phi + V'(\phi) = 0.</math>

===Dimensional analysis and scaling===
{{main|Natural units#"Natural units" (particle physics and cosmology)}}
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  {{mvar|t}}, with dimensions of time, can be readily converted into a ''length'', {{math|''l'' {{=}}''ct''}}, by using the [[velocity of light]],  {{mvar|c}}. Similarly, any length  {{mvar|l}} is equivalent to an inverse mass, {{mvar|ħ}}/''mc'', using [[Planck's constant]], {{mvar|ħ}}. In natural units, one thinks 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 [[Classical scaling dimension|mass dimension]] of the quantity. Knowing the dimensions of each quantity, allows one to ''uniquely restore'' conventional dimensions from a natural units expression in terms of this mass dimension, by simply reinserting the requisite powers of  {{mvar|ħ}} and {{mvar|c}} required for dimensional consistency.

One conceivable objection is that this theory is classical, and therefore it is not obvious how Planck's constant should be a part of the theory at all. If desired, one could indeed recast the theory without mass dimensions at all: However, this would be at the expense of slightly obscuring the connection with the quantum scalar field. Given that one has dimensions of mass, Planck's constant is thought of here as an essentially ''arbitrary fixed reference quantity of action'' (not necessarily connected to quantization), hence with dimensions appropriate to convert between mass and [[inverse length]].

====Scaling dimension====
The [[classical scaling dimension]], or mass dimension,  {{mvar|Δ}}, of  {{mvar|φ}} describes the transformation of the field under a rescaling of coordinates:
:<math>x\rightarrow\lambda x</math>
:<math>\phi\rightarrow\lambda^{-\Delta}\phi ~.</math>

The units of action are the same as the units of {{mvar|ħ}}, and so the action itself has zero mass dimension. This fixes the scaling dimension of the field {{mvar|φ}} to be
:<math>\Delta =\frac{D-2}{2}.</math>

====Scale invariance====
There is a specific sense in which some scalar field theories are [[Scale invariance|scale-invariant]]. While the actions above are all constructed to have zero mass dimension, not all actions are invariant under the scaling transformation
:<math>x\rightarrow\lambda x </math>
:<math>\phi\rightarrow\lambda^{-\Delta}\phi ~.</math>

The reason that not all actions are invariant is that one usually thinks of the parameters ''m'' and {{math|''g''<sub>''n''</sub>}}  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 {{math|''D''}} spacetime dimensions, the only dimensionless parameter  {{math|''g''<sub>''n''</sub>}} satisfies  {{mvar|n}} =  {{math|{{Fraction|2''D''|(''D'' − 2)}} }}. For example, in {{math|''D''}} = 4, only  {{math|''g''<sub>4</sub>}} is classically dimensionless, and so the only classically scale-invariant scalar field theory in {{math|''D''}} = 4 is the massless [[Quartic interaction|{{mvar|φ}}<sup>4</sup> theory]].

Classical scale invariance, however, normally does not imply quantum scale invariance, because of the [[renormalization group]] involved – see the discussion of the beta function below.

====Conformal invariance====
A transformation
:<math>x\rightarrow \tilde{x}(x)</math>
is said to be [[Conformal symmetry|conformal]] if the transformation satisfies
:<math>\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}</math>
for some function {{math|''λ''(''x'')}}.

The conformal group contains as subgroups the [[isometry|isometries]] of the metric <math>\eta_{\mu\nu}</math> (the [[Poincaré group]]) and also the scaling transformations (or [[Scale invariance|dilatation]]s) considered above. In fact, the scale-invariant theories in the previous section are also conformally-invariant.

==={{mvar|φ}}<sup>4</sup> theory===
{{See also|Quartic interaction}}
Massive {{mvar|φ}}<sup>4</sup> theory illustrates a number of interesting phenomena in scalar field theory.

The Lagrangian density is
:<math>\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.</math>

====Spontaneous symmetry breaking====
{{See also|Quartic interaction#Spontaneous symmetry breaking}}
This Lagrangian has a ℤ₂ symmetry under the transformation  {{math|''φ''→ −''φ''}}.
This is an example of an [[internal symmetry]], in contrast to a [[Spacetime symmetries|space-time symmetry]].

If {{math|''m''<sup>2</sup>}} is positive, the potential 
:<math>V(\phi)=\frac{1}{2}m^2\phi^2 +\frac{g}{4!}\phi^4</math> 
has a single minimum, at the origin. The solution ''φ''=0 is clearly invariant under the ℤ₂ symmetry.

Conversely, if {{math|''m''<sup>2</sup>}} is negative, then one can readily see that the potential 
:<math>\, V(\phi)=\frac{1}{2}m^2\phi^2+\frac{g}{4!}\phi^4\!</math> 
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 {{em|not}} invariant under the ℤ₂ symmetry of the action (in fact it maps each of the two vacua into the other). In this case, the ℤ₂ symmetry is said to be ''[[Spontaneous symmetry breaking|spontaneously broken]]''.

====Kink solutions====
The {{mvar|φ}}<sup>4</sup> theory with a negative {{mvar|m}}<sup>2</sup> also has a kink solution, which is a canonical example of a [[soliton]]. Such a solution is of the form
:<math>\phi(\vec{x}, t) = \pm\frac{m}{2\sqrt{\frac{g}{4!}}}\tanh\left[\frac{m(x - x_0)}{\sqrt{2}}\right]</math>
where  {{mvar|x}} is one of the spatial variables ({{mvar|φ}} is taken to be independent of  {{mvar|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
:<math>\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]</math>

This has a [[U(1)]], equivalently O(2) symmetry, whose action on the space of fields rotates <math>\phi\rightarrow e^{i\alpha}\phi</math>, for some real phase angle  {{mvar|α}}.

As for the real scalar field, spontaneous symmetry breaking is found if ''m''<sup>2</sup> is negative. This gives rise to Goldstone's [[Mexican hat potential]] which is a rotation of the double-well potential of a real scalar
field by 2π radians about the ''V''<math> (\phi) </math> axis. The symmetry breaking takes place in one higher dimension, i.e.  the choice of vacuum breaks a continuous ''U''(1) symmetry instead of a discrete one.
The two components of the scalar field are reconfigured as a massive mode and a massless [[Goldstone boson]].

===''O''(''N'') theory===

One can express the complex scalar field theory in terms of two real fields, ''φ''<sup>1</sup> = Re ''φ''   and    ''φ''<sup>2</sup> = Im ''φ'', 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 [[Orthogonal group|''O''(''N'')]] symmetry. The Lagrangian for an ''O''(''N'')-invariant scalar field theory is typically of the form
:<math>\mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi\cdot\partial_\nu\phi -V(\phi\cdot\phi)</math>
using an appropriate ''O''(''N'')-invariant [[inner product]].

==Quantum scalar field theory==
{{main|Canonical quantization#Real scalar field}}
A general reference for this section is Ramond, Pierre (2001-12-21). Field Theory: A Modern Primer (Second Edition). USA: Westview Press. {{isbn|0-201-30450-3}}, Ch. 4

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 quantum  [[Hamiltonian (quantum mechanics)|Hamiltonian]], a positive-definite operator which annihilates the vacuum.  A construction of a quantum scalar field theory is detailed in the [[canonical quantization]] article, which relies on canonical commutation relations among the fields.  Essentially, the infinity of classical oscillators repackaged in the scalar field as its (decoupled) normal modes, above, are now quantized in the standard manner, so the respective quantum operator field describes an infinity of [[quantum harmonic oscillator]]s acting on a respective [[Fock space]].

In brief, the basic variables are the quantum field {{mvar|φ}} and its canonical momentum {{mvar|π}}. Both these operator-valued fields are [[Hermitian operator|Hermitian]]. At spatial points {{vec|''x''}}, {{vec|''y''}} , at equal times, their [[canonical commutation relations]] are given by

:<math>\begin{align}
  \left[\phi\left(\vec{x}\right), \phi\left(\vec{y}\right)\right] =
    \left[\pi\left(\vec{x}\right), \pi\left(\vec{y}\right)\right] &= 0,\\
   \left[\phi\left(\vec{x}\right), \pi\left(\vec{y}\right)\right] &= i \delta\left(\vec{x} - \vec{y}\right),
\end{align}</math>

while the free [[Hamiltonian (quantum theory)|Hamiltonian]] is, similarly to above, 
:<math>H = \int d^3x \left[{1 \over 2}\pi^2 + {1 \over 2}(\nabla \phi)^2 + {m^2 \over 2}\phi^2\right].</math>

A spatial [[Fourier transform]] leads to [[momentum space]] fields
:<math>\begin{align}
  \widetilde{\phi}(\vec{k}) &= \int d^3x e^{-i\vec{k}\cdot\vec{x}}\phi(\vec{x}),\\
   \widetilde{\pi}(\vec{k}) &= \int d^3x e^{-i\vec{k}\cdot\vec{x}}\pi(\vec{x})
\end{align}</math>
which resolve to annihilation and creation operators
:<math>\begin{align}
           a(\vec{k}) &= \left(E\widetilde{\phi}(\vec{k}) + i\widetilde{\pi}(\vec{k})\right),\\
   a^\dagger(\vec{k}) &= \left(E\widetilde{\phi}(\vec{k}) - i\widetilde{\pi}(\vec{k})\right),
\end{align}</math>
where <math>E = \sqrt{k^2 + m^2}</math> .

These operators satisfy the commutation relations
:<math>\begin{align}
  \left[a(\vec{k}_1), a(\vec{k}_2)\right] =
    \left[a^\dagger(\vec{k}_1), a^\dagger(\vec{k}_2)\right] &= 0,\\
            \left[a(\vec{k}_1), a^\dagger(\vec{k}_2)\right] &= (2\pi)^3 2E \delta(\vec{k}_1 - \vec{k}_2).
\end{align}</math>

The state <math>| 0\rangle</math> annihilated by all of the operators ''a'' is identified as the ''bare vacuum'', and a particle with momentum  {{vec|''k''}}  is created by applying <math>a^\dagger(\vec{k})</math> to the vacuum.

Applying all possible combinations of creation operators to the vacuum constructs the relevant [[Hilbert space]]: This construction is called [[Fock space]].  The vacuum is annihilated by the Hamiltonian
:<math>H = \int {d^3k\over (2\pi)^3}\frac{1}{2} a^\dagger(\vec{k}) a(\vec{k}) , </math>
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 ''φ''<sup>4</sup> theory, this corresponds to adding a Wick ordered term ''g'':''φ''<sup>4</sup>:/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 (quantum mechanics)|perturbation theory]] by means of the [[Dyson series]], which gives the time-ordered products, or ''n''-particle Green's functions <math>\langle 0|\mathcal{T}\{\phi(x_1) \cdots \phi(x_n)\}|0\rangle</math> 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]].

{{see also|Quantum harmonic oscillator#Harmonic oscillators lattice: phonons}}

===Feynman path integral===
The [[Feynman diagram]] expansion may be obtained also from the Feynman [[path integral formulation]].<ref>A general reference for this section is {{cite book|last=Ramond|first=Pierre|title=Field Theory: A Modern Primer (Second Edition)|publisher=Westview Press|location=USA|date=2001-12-21|isbn=0-201-30450-3}}</ref> The [[time ordered]] [[vacuum expectation value]]s of polynomials in {{mvar|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,

:<math>\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^4x \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^4x \left({1 \over 2}\partial^\mu \phi \partial_\mu \phi - {m^2 \over 2}\phi^2 - {g \over 4!}\phi^4\right)}}.
</math>

All of these Green's functions may be obtained by expanding the exponential in ''J''(''x'')φ(''x'') in the generating function
:<math>
  Z[J] =
    \int \mathcal{D}\phi e^{i\int d^4x \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}{n!} J(x_1) \cdots J(x_n) \langle 0|\mathcal{T}\{\phi(x_1) \cdots \phi(x_n)\}|0\rangle.
</math>

A [[Wick rotation]] may be applied to make time imaginary. Changing the signature to (++++) then turns the Feynman integral into a [[partition function (statistical mechanics)|statistical mechanics partition function]] in [[Euclidean space]],
:<math>Z[J] = \int \mathcal{D}\phi e^{-\int d^4x \left[{1 \over 2}(\nabla\phi)^2 + {m^2 \over 2}\phi^2 + {g \over 4!}\phi^4 + J\phi\right]}.</math>

Normally, this is applied to the scattering of particles with fixed momenta, in which case, a [[Fourier transform]] is useful, giving instead
:<math>\widetilde{Z}[\widetilde{J}] = \int \mathcal{D}\widetilde\phi e^{-\int d^4p \left[{1 \over 2}\left(p^2 + m^2\right)\widetilde\phi^2 + {\lambda \over 4!}\widetilde\phi^4 - \widetilde{J}\widetilde\phi\right]}.</math>

The standard trick to evaluate this [[functional integral]] is to write it as a product of exponential factors, schematically,
:<math>\widetilde{Z}[\widetilde{J}]\sim\int \mathcal{D}\widetilde\phi \prod_p \left[e^{-{1 \over 2}\left(p^2 + m^2\right)\widetilde\phi^2} e^{-{g \over 4!}\widetilde\phi^4} e^{\widetilde{J}\widetilde\phi}\right].</math>
The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically through [[Feynman diagram]]s.

The integral with {{mvar|λ}} = 0 can be treated as a product of infinitely many elementary Gaussian integrals: the result may be expressed as a sum of [[Feynman diagrams]], calculated using the following Feynman rules:
* Each field  {{overset|~|''φ''}}(''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''<sup>''k''</sup>, all diagrams with ''n'' external lines and {{mvar|k}} vertices are constructed such that the momenta flowing into each vertex is zero. Each internal line is represented by a propagator 1/(''q''<sup>2</sup> + ''m''<sup>2</sup>), where {{mvar|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  {{overset|~|''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''/(''q''<sup>2</sup>−''m''<sup>2</sup>+''iε''), where the {{mvar|ε}} term represents the small Wick rotation needed to make the Minkowski-space Gaussian integral converge.

===Renormalization===
{{main|Beta function (physics)|Renormalization group}}
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.<ref>See the previous reference, or for more detail, {{cite book|last1=Itzykson|first1=Zuber|last2=Zuber|first2=Jean-Bernard|title=Quantum Field Theory|publisher=Dover|date=2006-02-24|isbn=0-07-032071-3}}</ref> 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 {{mvar|g}} on the scale {{mvar|λ}} is encoded by a [[Beta function (physics)|beta function]], {{math|''β''(''g'')}}, defined by
:<math>\beta(g) = \lambda\,\frac{\partial g}{\partial \lambda} ~.</math>

This dependence on the energy scale is known as "the running of the coupling parameter", and theory of this systematic 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 (quantum mechanics)|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 [[Feynman graph|loop]] contributions, due to the number of loops in the corresponding [[Feynman graph]]s).

The  {{math|''β''}}-function at one loop (the first perturbative contribution) for the {{mvar|φ}}<sup>4</sup> theory is
:<math>\beta(g) = \frac{3}{16\pi^2}g^2 + O\left(g^3\right) ~.</math>

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 persisted at large couplings, this would indicate the presence of a [[Landau pole]] at finite energy, arising from [[quantum triviality]].  However, the question can only be answered non-perturbatively, since it involves strong coupling.

A quantum field theory is said to be ''trivial'' when the renormalized coupling, computed through its [[Beta function (physics)|beta function]], goes to zero when the ultraviolet cutoff is removed. Consequently, the [[propagator]] becomes that of a free particle and the field is no longer interacting.

For a {{mvar|φ}}<sup>4</sup> interaction, [[Michael Aizenman]] proved that the theory is indeed trivial, for space-time dimension {{mvar|D}} ≥ 5.<ref name="Aiz81">
{{cite journal
 |last=Aizenman|first=M. |authorlink=Michael Aizenman
 |year=1981
 |title=Proof of the Triviality of ''ϕ{{su|b=d|p=4}}'' Field Theory and Some Mean-Field Features of Ising Models for ''d'' > 4
 |journal=[[Physical Review Letters]]
 |volume=47
 |issue=1 |pages=1–4 |doi=10.1103/PhysRevLett.47.1 |bibcode=1981PhRvL..47....1A
}}</ref>

For {{mvar|D}} = 4, the triviality has yet to be proven rigorously, but [[Quantum triviality|lattice computations]] have provided strong evidence for this.  This fact is important as [[quantum triviality]] can be used to bound or even ''predict'' parameters such as the [[Higgs boson]] mass.  This can also lead to a predictable Higgs mass in [[Physics applications of asymptotically safe gravity#The mass of the Higgs boson|asymptotic safety]] scenarios.<ref name="TrivPurs">{{cite journal| authorlink=David J E Callaway | first=D. J. E. |last=Callaway | year=1988
| title=Triviality Pursuit: Can Elementary Scalar Particles Exist? | journal=[[Physics Reports]]
|volume=167 | issue=5 | pages=241–320 | doi=10.1016/0370-1573(88)90008-7
|bibcode = 1988PhR...167..241C }}</ref>

==See also==

*[[Renormalization]]
*[[Quantum triviality]]
*[[Landau pole]]
*[[Scale invariance#CFT description]]
*[[Scalar electrodynamics]]

== Notes ==
{{Reflist|2}}

== References ==
* {{cite book|last1=Peskin|first1=M.|last2=Schroeder|first2=D.|authorlink1=Michael Peskin|title=An Introduction to Quantum Field Theory|publisher=Westview Press|year=1995|isbn=978-0201503975|url=https://books.google.com/books?id=EVeNNcslvX0C&printsec=frontcover&hl=sv#v=onepage&q&f=false}}
* {{cite book|last=Weinberg|first=S.|authorlink=Steven Weinberg|title=The Quantum Theory of Fields|publisher=Cambridge University Press|year=1995|volume=I|isbn=0 521 55001 7|url-access=registration|url=https://archive.org/details/quantumtheoryoff00stev}}
* {{cite book|last=Weinberg|first=S.|title=The Quantum Theory of Fields|publisher=Cambridge University Press|year=1998|volume=II|isbn=0 521 55002 5}}
* {{cite book|last=Srednicki|first=M.|title=Quantum Field Theory|publisher=Cambridge University Press|year=2007|isbn=9780521864497}}
* {{cite book|last=Zinn-Justin|first=J|authorlink=Jean Zinn-Justin|title=Quantum Field Theory and Critical Phenomena|publisher=Oxford University Press|year=2002|isbn=978-0198509233}}

==External links==
*[http://www.phys.uu.nl/~thooft/lectures/basisqft.pdf The Conceptual Basis of 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.

[[Category:Quantum field theory]]
[[Category:Mathematical physics]]
[[Category:Theoretical physics]]