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\psi(t)\rang, where m is the particle'smass, V is the applied potential, and \psi\rang denotes the quantum state (we are using braket notation). We wish to consider how this problem generalizes to N particles. There are two motivations for studying the manyparticle problem. The first is a straightforward need in condensed matter physics, where typically the number of particles is on the order of Avogadro's number (6.0221415 x 10^{23}). The second motivation for the manyparticle problem arises from particle physics and the desire to incorporate the effects of special relativity. If one attempts to include the relativistic rest energy into the above equation, the result is either the KleinGordon 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. It turns out that such inconsistencies arise from neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous massenergy 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. Thus, a consistent relativistic quantum theory must be formulated as a manyparticle theory. Furthermore, we will assume that the N particles are indistinguishable. As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multiparticle states are rather complicated to write. For example, the general quantum state of a system of N bosons is written as
where \phi_i\rang are the singleparticle states, N_j is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. The way to simplify this problem is to turn it into a quantum field theory. Second quantizationIn this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identicalparticle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral, which uses a Lagrangian formulation. For an overview, see the article on quantization.Second quantization of bosonsFor simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal singleparticle states by \phi_1\rang, \phi_2\rang, \phi_3\rang, and so on. For example, the 3particle state with one particle in state \phi_1\rang and two in state\phi_2\rang is

\phi_2\rang \phi_1\rang \phi_2\rang + \phi_2\rang 
\phi_1\rang \right]. The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the singleparticle states \phi_1\rang, \phi_2\rang, etc. This is simply another way of labelling the states. For instance, the above 3particle state is denoted as
The next step is to expand the Nparticle state space to include the state spaces for all possible values of N. This extended state space, known as a Fock space, is composed of the state space of a system with no particles (the socalled vacuum state), plus the state space of a 1particle system, plus the state space of a 2particle system, and so forth. It is easy to see that there is a onetoone correspondence between the occupation number representation and valid boson states in the Fock space. At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number j\cdots, indicating which of the singleparticle states \phi_1\rang, \phi_2\rang, \cdots\phi_j\rang\cdots it refers to. The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles. They are analogous to "ladder operators" in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator a_2 and creation operator a_2^\dagger have the following effects:
It can be shown that these are operators in the usual quantum mechanical sense, i.e. linear operators acting on the Fock space. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation \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}, where \delta 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 singleparticle state. Adding or removing bosons from each state is therefore analogous to exciting or deexciting a quantum of energy in a harmonic oscillator. The Hamiltonian of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, the Hamiltonian of a field of free (noninteracting) bosons is
where E_k is the energy of the kth singleparticle energy eigenstate. Note that
Second quantization of fermionsIt turns out that a different definition of creation and annihilation must be used for describing fermions. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers N_i can only take on the value 0 or 1. The fermionic annihilation operators c and creation operators c^\dagger are defined by their actions on a Fock state thus
These obey an anticommutation relation: \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} One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share singleparticle states, in accordance with the exclusion principle. Field operatorsWe have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the singleparticle quantum states. However, as we have discussed, it is more natural to think about a "field", such as the electromagnetic field, as a set of degrees of freedom indexed by position.To this end, we can define field operators that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity. Singleparticle 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 \phi(\mathbf{r}) is
The bosonic field operators obey the commutation relation \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'}) where \delta(x) 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 singleparticle 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
where the indices i and j run over all particles, then the field theory Hamiltonian is
This looks remarkably like an expression for the expectation value of the energy, with \phi playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from spaceprojected Hamiltonians. Implications of quantum field theoryUnification of fields and particlesThe "second quantization" procedure that we have outlined in the previous section takes a set of singleparticle quantum states as a starting point. Sometimes, it is impossible to define such singleparticle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible (for various reasons) to define a wavefunction for a single photon. In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory. The quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as welldefined creation and annihilation operators obeying commutation or anticommutation relations.Quantum field theory thus provides a unified framework for describing "fieldlike" objects (such as the electromagnetic field, whose excitations are photons) and "particlelike" objects (such as electrons, which are treated as excitations of an underlying electron field). Physical meaning of particle indistinguishabilityThe second quantization procedure relies crucially on the particles being identical. We would not have been able to construct a quantum field theory from a distinguishable manyparticle system, because there would have been no way of separating and indexing the degrees of freedom.Many physicists prefer to take the converse interpretation, which is that quantum field theory explains what identical particles are. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental. Particle conservation and nonconservationDuring second quantization, we started with a Hamiltonian and state space describing a fixed number of particles (N), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations N is an important and perfectly welldefined quantity, e.g. if we are describing a gas of atoms sealed in a box. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator \hat{N}, which measures the total number of particles present. As with any quantum mechanical observable, \hat{N} is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the Nparticle subspace of the total Fock space, and the situation could equally well be described by ordinary Nparticle quantum mechanics.For example, we can see that the freeboson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator a_k is immediately put back by the creation operator a_k^\dagger. On the other hand, it is possible, and indeed common, to encounter quantum states that are not eigenstates of \hat{N}, which do not have welldefined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as quantum superpositions of states having different values of N. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as
where a_k^\dagger and a_k denotes the bosonic creation and annihilation operators, c_k^\dagger and c_k denotes the fermionic creation and annihilation operators, and V_q is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k+q. (In fact, this type of Hamiltonian is used to describe interaction between conduction electrons and phonons in metals. The interaction between electrons and photons is treated in a similar way, but 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 typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case. In condensed matter physics, states with illdefined particle numbers are particularly 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. Axiomatic approachesThe preceding description of quantum field theory follows the spirit in which most physicists approach the subject. However, it is not mathematically rigorous. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes.The first class of axioms, first proposed during the 1950s, include the Wightman, OsterwalderSchrader, and HaagKastler systems. They attempted to formalize the physicists' notion of an "operatorvalued field" within the context of functional analysis, and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the spinstatistics theorem and the CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to lowdimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of constructive quantum field theory. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others. During the 1980s, a second set of axioms based on geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with Michael Atiyah and Graeme Segal, and was notably expanded upon by Edward Witten, Richard Borcherds, and Maxim Kontsevich. However, most physicallyrelevant quantum field theories, such as the Standard Model, are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in representation theory, algebraic topology, and differential geometry. Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in YangMills theory—is linked to this issue. Phenomena associated with quantum field theoryIn the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.RenormalizationEarly in the history of quantum field theory, it was found that manyseemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances which each give a finite contribution.Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field which it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere. The solution to the problem, presciently suggested by Stueckelberg, independently by Bethe after the crucial experiment by Lamb, implemented at one loop by Schwinger, and systematically extended to all loops by Feynman and Dyson, with converging work by Tomonaga in isolated postwar Japan, is called renormalization. The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a cutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by 't Hooft and Veltman, is a symmetry preserving cutoff for perturbation theory. There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice. The rule is that one computes physical quantities in terms of the observable parameters such as the physical mass, not the bare parameters such as the bare mass. The main point is not that of getting finite quantities (any regularization proceduredoes that), but to eliminate the regularization parameters by a suitable addition of counterterms to the original Lagrangian.The main requirements on the counterterms are a) Locality (polynomials in the fields and their derivatives) and b) Finiteness (number of monomials in the Lagrangian that remain finite after the introduction of all the necessary counterterms). The reason for (b) is that each new counterterm leaves behind a free parameter of the theory (like physical mass). There is no way such a parameter can be fixed other than by its experimental value, so one gets not a single theory but a family of theories parameterized by as many free parameters as the counterterms added to the Lagrangian. Since a theory with an infinite number of free parameters has virtually no predictive power the finiteness of the number of counterterms is required. On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physicallyobservable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit. The renormalization procedure only works for a certain class of quantum field theories, called renormalizable quantum field theories. A theory is perturbatively renormalizable when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined nonperturbatively, the problems only show up at distance scales which are exponentially small in the inverse coupling for weak couplings. The Standard Model of particle physics is perturbatively renormalizable, and so are its component theories (quantum electrodynamics/electroweak theory and quantum chromodynamics). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the asymptotically free SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined. The renormalization group describes how renormalizable theories emerge as the long distance lowenergy effective field theory for any given highenergy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying highenergy shortdistance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants. Gauge freedomA gauge theory is a theory that admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical. Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is  one may shift the phase of all wave functions so that the shift may be different at every point in spacetime. This is a local symmetry. However, in order for a welldefined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In quantum electrodynamics this gauge field is the electromagnetic field. The change of local gauge of variables is termed gauge transformation.In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics. The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "nonphysical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is nonconsistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non unitary and again inconsistent (see optical theorem). In general, the gauge transformations of a theory consist several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group dimension (i.e. number of generators forming a basis). All the fundamental interactions in nature are described by gauge theories. These are:
SupersymmetrySupersymmetry assumes that every fundamental fermion has a superpartner that is a boson and vice versa. It was introduced in order to solve the socalled Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory.The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite. Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a socalled soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider. See also
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