For this reason, scalar field theories are often used for purposes of introduction of novel concepts and techniques.
The corresponding Euler–Lagrange equation of motion is now 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 readily converted into a length, l =ct, by using the velocity of light, c. Similarly, any length l is equivalent to an inverse mass, ħ=lmc, using the Planck constant, ħ.
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, the Planck 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.
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.
For a scalar field theory with D spacetime dimensions, the only dimensionless parameter gn satisfies n = 2D⁄(D − 2) .
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
As for the real scalar field, spontaneous symmetry breaking is found if m2 is negative.
The two components of the scalar field are reconfigured as a massive mode and a massless Goldstone boson.
This can be generalised to a theory of N scalar fields transforming in the vector representation of the O(N) symmetry.
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 oscillators acting on a respective Fock space.
In brief, the basic variables are the quantum field φ and its canonical momentum π.
At spatial points x→, y→ and at equal times, their canonical commutation relations are given by while the free Hamiltonian is, similarly to above, A spatial Fourier transform leads to momentum space fields which resolve to annihilation and creation operators where
annihilated by all of the operators a is identified as the bare vacuum, and a particle with momentum k→ is created by applying
The vacuum is annihilated by the Hamiltonian where the zero-point energy has been removed by Wick ordering.
These are constructed in perturbation theory by means of the Dyson series, which gives the time-ordered products, or n-particle Green's functions
[4] 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, All of these Green's functions may be obtained by expanding the exponential in J(x)φ(x) in the generating function 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, Normally, this is applied to the scattering of particles with fixed momenta, in which case, a Fourier transform is useful, giving instead where
The standard trick to evaluate this functional integral is to write it as a product of exponential factors, schematically, The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically through Feynman diagrams of the Quartic interaction.
The integral with g = 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: The last rule takes into account the effect of dividing by ~Z[0].
[5] 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 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, where one assumes that the coupling constant is small.
The β-function at one loop (the first perturbative contribution) for the φ4 theory is 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.
A quantum field theory is said to be trivial when the renormalized coupling, computed through its beta function, goes to zero when the ultraviolet cutoff is removed.
For a φ4 interaction, Michael Aizenman proved that the theory is indeed trivial, for space-time dimension D ≥ 5.
[6] For D = 4, the triviality has yet to be proven rigorously, but 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.