Quantum field theory
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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 which to formulate consistently quantum theories of many-particle systems, especially in situations where particles may be created and destroyed. Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.
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Why quantum field theory
Quantum field theory was necessary in order to compute the power radiated by an atom when it dropped from one quantum state to another of lower energy. This problem was first examined by Max Born and Pascual Jordan in 1925. Quantum field theory was necessiated in the first place for a quantum treatment of the only known classical field, ie, electromagnetism. In 1926, Max Born, Werner Heisenberg and Pascual Jordan wrote down the quantum theory of the electromagnetic field neglecting polarization and sources to obtain what would today be called a free field theory. In order to quantize this theory, they used the canonical quantization procedure. In 1927 Paul Dirac gave the first consistent treatment of this problem. So, in the first place, quantum field theory was necessiated by the need to treat a situation where the number of particles changes— here one atom in the initial state becomes an atom and a photon in the final state.
It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. Jordan and Wolfgang Pauli showed in 1928 that commutators of the field were actually Lorentz invariant. By 1933, Niels Bohr and Leon Rosenfeld had related these commutation relations to a limitation on the ability to measure fields at space-like separation. The development of the Dirac equation and the hole theory drove quantum field theory to explain these using the ideas of causality in relativity, work that was completed by Wendell Furry and Robert Oppenheimer using methods developed for this purpose by Vladimir Fock. This need to put together relativity and quantum mechanics was a second motivation which drove the development of quantum field theory. This thread was crucial to the eventual development of particle physics and the modern (partially) unified theory of forces called the standard model
In 1927 Jordan tried to extend the canonical quantization of fields to the wave function which appeared in the quantum mechanics of particles, giving rise to the equivalent name second quantization for this procedure. In 1928 Jordan and Eugene Wigner found that the Pauli exclusion principle demanded that the electron field be expanded using anti-commuting creation and annihilation operators. This was the third thread in the development of quantum field theory— the need to handle the statistics of multi-particle systems consistently and with ease. This thread of development was incorportated into many-body theory, and strongly influenced condensed matter physics and nuclear physics.
Technical statement
Quantum field theory corrects several limitations of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equation, in its most commonly encountered form, is
- <math> \left[ \frac{|\mathbf{p}|^2}{2m} + V(\mathbf{r}) \right]
|\psi(t)\rang = i \hbar \frac{\partial}{\partial t} |\psi(t)\rang <math>
where <math>|\psi\rang<math> denotes the quantum state (nota) of a particle with mass <math>m<math>, in the presence of a potential <math>V<math>.
The first problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of <math>N<math> bosons is written as
- <math> |\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j!}{N!}} \sum_{p} |\phi_{p(1)}\rang \cdots |\phi_{p(N)} \rang <math>
where <math>|\phi_i\rang<math> are the single-particle states, <math>N_j<math> is the number of particles occupying state <math>j<math>, and the sum is taken over all possible permutations <math>p<math> acting on <math>N<math> elements. In general, this is a sum of <math>N!<math> (<math>N<math> factorial) distinct terms, which quickly becomes unmanageable as <math>N<math> increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is the Avogadro's number, approximately 1023.
The second problem arose when trying to reconcile the Schrödinger equation with special relativity. It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a 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. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory.
Quantizing a classical field theory
Second quantization
Quantum field theory solves these problems by consistently quantizing a theory of many particles. It does this by introducing a field as a quantum operator whose eigenstates are particles.
Two caveats should be made before proceeding further:
- Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime.
- Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, particles). Quantum field theory is the quantum mechanics of this infinite system.
The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.
Second quantization for bosons
Suppose we have a system of <math>N<math> bosons which can occupy mutually orthogonal single-particle states <math>|\phi_1\rang, |\phi_2\rang, |\phi_3\rang,<math> and so on. The usual method of writing a multi-particle state 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 <math>N!<math> terms. In contrast, in the second quantized approach we will 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 specific, suppose that <math>N=3<math>, with one particle in state <math>|\phi_1\rang<math> and two in state<math>|\phi_2\rang<math>. The normal way of writing the wavefunction is
- <math> \frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang
|\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang |\phi_2\rang |\phi_1\rang \right] <math>
In second quantized form, we write this as
- <math> |1, 2, 0, 0, 0, \cdots \rangle <math>
which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."
Though the difference is entirely notational, the latter form makes it easy for us 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. The bosonic annihilation operator <math>a_2<math> and creation operator <math>a_2^\dagger<math> have the following effects:
- <math> a_2 | N_1, N_2, N_3, \cdots \rangle = \sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \cdots \rangle <math>
- <math> a_2^\dagger | N_1, N_2, N_3, \cdots \rangle = \sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \cdots \rangle <math>
We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them.
The bosonic creation and annihilation operators obey the commutation relation
- <math>
\left[a_i , a_j \right] = 0 \quad,\quad \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad \left[a_i , a_j^\dagger \right] = \delta_{ij} <math>
where <math>\delta<math> stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.
The final step toward obtaining a quantum field theory is to re-write our original <math>N<math>-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is
- <math>H = \sum_k E_k \, a^\dagger_k \,a_k<math>
where <math>E_k<math> is the energy of the <math>k<math>-th single-particle energy eigenstate. Note that
- <math>a_k^\dagger\,a_k|\cdots\rangle=N_k| \cdots, N_k, \cdots \rangle<math>.
Second quantization for fermions
It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers <math>N_i<math> can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators <math>c<math> and creation operators <math>c^\dagger<math> by
- <math> c_j | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = 0 <math>
- <math> c_j | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \cdots, N_j = 0, \cdots \rangle <math>
- <math> c_j^\dagger | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \cdots, N_j = 1, \cdots \rangle <math>
- <math> c_j^\dagger | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = 0 <math>
The fermionic creation and annihilation operators obey an anticommutation relation,
- <math>
\left\{c_i , c_j \right\} = 0 \quad,\quad \left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad \left\{c_i , c_j^\dagger \right\} = \delta_{ij} <math>
One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.
Significance of creation and annihilation operators
When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol <math>N<math>, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations <math>N<math> is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator <math>a_k<math> destroys a particle during an infinitesimal time step, the creation operator <math>a_k^\dagger<math> to the left of it instantly puts it back. Therefore, if we start with a state of <math>N<math> non-interacting particles then we will always have <math>N<math> particles at a later time.
On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of <math>N<math>. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by <math>c_k^\dagger<math> and <math>c_k<math>, we could add a "potential energy" term to our Hamiltonian such as:
- <math>V = \sum_{k,q} V_q (a_q + a_{-q}^\dagger) c_{k+q}^\dagger c_k <math>
This describes processes in which a fermion in state <math>k<math> either absorbs or emits a boson, thereby being kicked into a different eigenstate <math>k+q<math>. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case.
In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.
Field operators
We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator <math>\phi(\mathbf{r})<math> is
- <math>\phi(\mathbf{r}) \equiv \sum_{i} e^{i\mathbf{k}_i\cdot \mathbf{r}} a_{i} <math>
The bosonic field operators obey the commutation relation
- <math>
\left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'}) <math>
where <math>\delta(x)<math> stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say
- <math>H = - \frac{\hbar^2}{2m} \sum_i \nabla_i^2 + \sum_{i < j} U(|\mathbf{r}_i - \mathbf{r}_j|) <math>
where the indices <math>i<math> and <math>j<math> run over all particles, then the field theory Hamiltonian is
- <math>H = - \frac{\hbar^2}{2m} \int d^3\!r \; \phi(\mathbf{r})^\dagger \nabla^2 \phi(\mathbf{r}) + \int\!d^3\!r \int\!d^3\!r' \; \phi(\mathbf{r})^\dagger \phi(\mathbf{r}')^\dagger U(|\mathbf{r} - \mathbf{r}'|) \phi(\mathbf{r'}) \phi(\mathbf{r}) <math>
This looks remarkably like an expression for the expectation value of the energy, with <math>\phi<math> playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.
Quantization of classical fields
So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon, so a quantum field theory must be formulated right from the start.
The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of <math>N<math> particles, there are <math>3N<math> coordinate variables corresponding to the position of each particle, and <math>3N<math> conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as
- <math>\left[ q_i , p_j \right] = \delta_{ij}<math>
For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential <math>\phi(\mathbf{x})<math> and the vector potential <math>\mathbf{A}(\mathbf{x})<math> at every point <math>\mathbf{x}<math>. This is an uncountable set of variables, because <math>\mathbf{x}<math> is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory.
Path integral methods
The axiomatic approach
There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. The most prominent of these are the Wightman axioms and the Haag-Kastler axioms.
The classic results gained from the axiomatic approach are the PCT Theorem (stating that the combination of parity, time and charge inversion is an unbroken symmetry) and the spin-statistics theorem (stating that particles of integer valued spin follow the Bose-Einstein statistics and particles of half-integer spin follow the Fermi statistics).
Renormalization
The essence of quantum field theory is renormalization. A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture—
- <math>|\psi(t)\rangle = e^{iH_It} |\psi(0)\rangle = \left[1+iH_It-\frac12 H_I^2t^2 -\frac i{3!}H_I^3t^3 + \frac1{4!}H_I^4t^4 + \cdots\right] |\psi(0)\rangle.<math>
Taking the overlap with the intial state, one retains the even powers of HI. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave-function renormalization and mass renormalization. Similiar corrections to the interaction Hamiltonian, HI, include vertex renormalization, or, in modern language, effective field theory.
Gauge theories
Supersymmetry
Beyond local field theory
History
More details can be found in the article on the history of quantum field theory.
Quantum field theory was created by Dirac when he attempted to quantize the electromagnetic field in the late 1920s. The early development of the field involved Fock, Pauli, Heisenberg, Bethe, Tomonaga, Schwinger, Feynman, and Dyson. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.
Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed by the 1970s through the word of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer.
Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s which led to the seminal reformulation of quantum field theory by Kenneth Wilson.
The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics.
See also
- Examples of quantum field theory models
- Feynman path integral
- Quantum chromodynamics
- Quantum electrodynamics
- Schwinger-Dyson equation
Suggested reading
- Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 1983), [ISBN 0198511558]
- A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press)
- M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0201503972]
- Weinberg, Steven, The Quantum Theory of Fields (3 volumes)
- Siegel, Warren, Fields (http://insti.physics.sunysb.edu/%7Esiegel/Fields2.pdf) (also available from arXiv:hep-th/9912205)
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