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Quantum field theory

Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework widely used in particle physics and condensed matter physics. In particular, the quantum theory of the electromagnetic field, known as quantum electrodynamics, is one of the most well-tested and successful theories in physics. The fundamentals of quantum field theory were developed during 19351955, notably by Dirac, Pauli, Tomonaga, Schwinger[?], Feynman, and Dyson.

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Shortcomings of ordinary quantum mechanics

Quantum field theory corrects several deficiencies of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equation, in its most commonly-encountered form, is

\left[ - \frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r}, t) = i \hbar \frac{\partial \Psi}{\partial t} (\mathbf{r}, t) </math>

where Ψ is the wavefunction of a particle, m its mass, and V an applied potential energy.

There are two problems with this equation. Firstly, it is not relativistic, reducing to classical mechanics rather than relativistic mechanics in the correspondence limit. To see this, we note that the first term on the left is only the classical kinetic energy p²/2m, with the rest energy mc² omitted. It is possible to modify the Schrödinger equation to include the rest energy, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy spectra[?] which extend to -∞, so that there is no ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory.

The second problem occurs when we seek to extend the equation to large numbers of particles. It was discovered that quantum mechanical particles of the same species are indistinguishable, in the sense that the wavefunction of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its consituent particles are exchanged. This makes the wavefunction of systems of many particles extremely complicated. For example, the general wavefunction of a system of N bosons is written as

\Phi(r_1, ..., r_N) = \frac{1}{\sqrt{N!}} \sum_{p} \phi_{p(1)} (r_1) \cdots \phi_{p(N)} (r_N) </math>

where ri are the coordinates of the i-th particle, φi are the single-particle wavefunctions, and the sum is taken over all possible permutations of p elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases.

Quantum fields

Both of the above problems are resolved by moving our attention from a set of indestructible particles to a quantum field. The procedure by which quantum fields are constructed from individual particles was introduced by Dirac, and is (for historical reasons) known as second quantization[?].

We should mention two possible points of confusion. Firstly, the aforementioned "field" and "particle" descriptions do not refer to wave-particle duality. By "particle", we refer to entities which possess both wave and point-particle properties in the usual quantum mechanical sense; for example, these "particles" are generally not located at a fixed point, but have a certain probability of being found at each position in space. What we refer to as a "field" is an entity existing at every point in space, which regulates the creation and annihilation of the particles. Secondly, quantum field theory is essentially quantum mechanics, and not a replacement for quantum mechanics. Like any quantum system, a quantum field possesses a Hamiltonian H (albeit one that is more complicated than typical single-particle Hamiltonians), and obeys the usual Schrödinger equation

<math> \mathbf{H} \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle</math>

(Quantum field theory is often formulated in terms of a Lagrangian, which is more convenient to work with. However, the Lagrangian and Hamiltonian formulations are entirely equivalent.)

In second quantization, we make use of particle indistinguishability by specifying multi-particle wavefunctions in terms of single-particle occupation numbers. For example, suppose we have a system of N bosons which can occupy various single-particle states φ1, φ2, φ3, and so on. The usual method of writing a multi-particle wavefunction is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In the second quantized approach, we simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be precise, suppose that N = 3, with one particle in state φ1 and two in state φ2. The normal way of writing the wavefunction is

\frac{1}{\sqrt{3}} \left[ \phi_1(r_1) \phi_2(r_2) \phi_2(r_3) + \phi_2(r_1) \phi_1(r_2) \phi_2(r_3) + \phi_2(r_1) \phi_2(r_2) \phi_1(r_3) \right] </math>

whereas in second quantized form it is simply

<math> |1, 2, 0, 0, \cdots \rangle </math>

Though the difference is entirely notational, the latter form makes it extremely easy to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. For example, the annihilation operator a2 has the following effects:

<math> a_2 | 1, 2, 0, 0, \cdots \rangle \equiv | 1, 1, 0, 0, \cdots \rangle \sqrt{2} </math>
<math> a_2 | 1, 1, 0, 0, \cdots \rangle \equiv | 1, 0, 0, 0, \cdots \rangle </math>
<math> a_2 | 1, 0, 0, 0, \cdots \rangle \equiv \quad 0 </math>

(The √2 factor in the first line normalizes the wavefunction, and is not important.)

Finally, we introduce field operators that define the probability of creating or destroying a particle at a particular point in space. It turns out that single-particle wavefunction are usually enumerated in terms of their momenta (as in the particle in a box problem), so field operators can be constructed by applying the Fourier transform to the creation and annihilation operators. For example, the bosonic field annihilation operator φ(r) (which is not to be confused with the wavefunction) is

<math>\phi(\mathbf{r}) \equiv \sum_{i} e^{i\mathbf{k}_i\cdot \mathbf{r}} a_{i} </math>

In quantum field theories, Hamiltonians are written in terms of either the creation and annihilation operators or, equivalently, the field operators. The former practice is more common in condensed matter physics, whereas the latter is more common in particle physics since it makes it easier to deal with relativity. An example of a Hamiltonian written in terms of creation and annihilation operators is

<math>\mathbf{H} = \sum_k E_k \, a^\dagger_k \,a_k</math>

This describes a field of free (non-interacting) bosons, where Ek is the kinetic energy of the k-th momentum mode. In fact, this Hamiltonian is useful for describing non-interacting phonons.

Wightman axioms

W0 (assumptions of relativistic quantum mechanics)

The pure states are given by the vectors of some separable complex Hilbert space on which continuous unitary representation U(a,L) of Poincaré group acts according to the following interpretation:

An ensamble corresponding to U(a,L)|v> is to be described with respect to the coordinates <math>x'=L^{-1}(x-a)</math> in exactly the same way as an ensemble corresponding to |v> is to be described with respect to the coordinates x.

The representation U(a,L) fulfills the spectral condition - that eigenvalues of energy-momentum are contained in the forward cone. There is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincare group. It is called a vacuum.

W1 (assumptions on the domain and continuity of the field)

For each test function f, there exists a set of operators <math>A_1(f),\ldots A_n(f)</math> which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing vacuum. The fields A are operator valued tempered distributions. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).

W2 (transformation law of the field)

The fields are covariant under the action of Poincaré group, and they transform according to some representation S of the Lorentz group: <math>U(a,L)^{\dagger}A(x)U(a,L)=S(L)A(L^{-1}(x-a))</math>

W3 (local commutativity or microscopic causality)

If the supports of two fields are space-like separated, then the fields either commute or anticommute

Cyclicity of a vacuum, and uniqueness of a vacuum are sometimes considered separately. Also, there is property of asymptotic completeness - that hilbert state space is spanned by the asymptotic spaces <math>H^{in}</math> and <math>H^{out}</math>, appearing in the collision S matrix[?]. The other important property of field theory is mass gap[?] which is not required by the axioms - that energy-momentum spectrum has a gap between zero and some positive number.

From these axioms, certain general theorems follow:

  • Connection between spin and statistic - fields which transform according to half integer spin anticommute, while those with integer spin commute (axiom W3)
  • PCT theorem - there is general symmetry under change of parity, particle-antiparticle reversal and time inversion (none of these symmetries alone exists in nature, as it turns out)

Arthur Wightman showed that the vacuum expectation value distributions, satisfying certain set of properties which follow from the axioms, are sufficient to reconstruct the field theory - Wightman reconstruction theorem[?], although he did not get the uniqueness of a vaccuum.

If the theory has a mass gap[?], i.e. there are no masses between 0 and some constant greater than zero, then vacuum expectation distributions are asimptotically independent in distant regions.

Haag's theorem[?] says that there can be no interaction picture - that we cannot use the Fock space of noninteracting particles as a Hilbert space - in the sense that we would identify Hilbert spaces via field polynomials acting on a vacuum at a certain time.

Currently, there is no proof that these axioms can be satisfied for gauge theories in dimension 4 - Standard model thus has no strict foundations. There is a million dollar prize for a proof that these axioms can be satisfied for gauge theories[?], with the additional requirement of a mass gap.

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