Fermions and Bosons
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[edit] Fermions and Bosons; The Janus of Quantum Theory
[The following is an extract from my book which can be accessed from http://www.lulu.com/content/72115] Paul Bennett
In Newtonian mechanics objects are described classically as particles in which the position and momentum at a particular instant can be specified exactly. A collection of particles in thermal equilibrium is described in terms of Boltzmann statistics, in which particles are distinguishable. In quantum mechanics a 'particle' is described by a wavefunction which is acted upon by hermitian operators which extracts eigenvalues that represent measurable observables. Such observables as momentum and position although complimentary are also mutually exclusive (as enshrined by Heisenberg’s uncertainty principle). The revolution that is quantum theory can be summerised as a transition from functions to operators and from classical approach of Poison brackets to the quantum commutators (involving Planks constant 'h' and the imaginary number 'i' ). A wavefunction does however have an ‘internal’ structure known as a spinor which is crucial in determining how the particle behaves. A collection of such quanta is described either by Einstein-Bose statistics (in the case of integer spin bosons) or by Fermi-Dirac statistics (half odd-integer spin fermions), both of which involve ‘particles’ that are not distinguishable. It was in fact Dirac who first introduced the concept of spinors into QT as a means of making Schroedingers equation compatible with SR (such quanta obeyed the spin statistics of an assembly of fermions).
Hence in QT all entities can be classified according to whether they obey such fermionic statistics (with its anti-symmetric wavefunction) or bosonic (symmetric) statistics. There is a whole lot of group theory which shows that half-integer spins obey the former (e.g. those whose angular momentum is 1/2h). This in turn means that 2 fermions cannot occupy the same quantum state (Pauli’s Exclusion Principle), an important fact since it explains the stability of the electron configurations in an atom and hence allows the richness of the periodic table. Bosons on the other hand like to be in the same quantum state and the fact that they can be stimulated to do so is exploited by the 'Light Amplification by Stimulated Emission of Radiation' (LASER). Super cooled helium-4 also obeys these statistics (in this case bosons are conserved) and thus allows the production of an Einstein-Bose condensate, the so- called fourth state of matter (what ever happened to plasma?).Whole integer spinors (bosons) faithfully represent the rotational group (they behave like the squares in a Rubic cube) while fermions exhibit a non-faithful representation, hence if you rotate an electron by one revolution it ends up being 'upside-down'.[ This property is responsible for its 'anti-social' behaviour, since it gives rise to the anti-symmetric wavefunction that characterises an assembly of fermions.]
Let us take another look at the origin of this double-headed aspect of QT. A physical law needs to be invariant under rotation, hence we need to know how to express a transformation, so as to maintain the symmetry of an equation, with respect to all observers who are oriented in different directions. Wavefunctions (state vectors) exist in Hilbert space which involve complex numbers and undergo unitary transformations in order that probabilities are conserved. The special unitary group SU(2) achieves this and acts on the 2-dimensional space of spinors (so called because it represents spin states). When the XYZ axes are rotated by an angle @ (e.g. 360) the spinor rotates by only @ /2 (i.e.180). Hence we have two important representations of the rotational groups viz. O(3) acting on real vectors in Euclidean 3-D space and SU(2) acting on complex space. The later is actually simpler (simply connected) while the latter is doubly connected.
Wavefunctions must transform under this (compact) unitary group in order to preserve transition amplitudes however relativity transformations involves a non-compact Lie group. For simple rotational transformations this can be achieved as already mentioned, by using a spinor representation, which under the action of the Lorentz group, undergoes a unimodular transformation and under the restriction to the rotational sub-group, it also undergoes a unitary change. The Pauli matrices generate such a special unitary group SU(2). Hence if an observer’s axes are rotated, the group that transforms this wavefunction must be both unimodular and unitary. This requires using spinor space not vector space i.e. as vector co-ordinates go through an Euclidean rotation, the spinors wavefunction evolves in a unitary manner.
This spinor space is not however a faithful representation of the rotational group and leads to the result that 2 rotations are needed to make the spinor function return to its original state and it is this which is the cause of the Lande’ factor g = 2. [One rotation induces a negative sign producing an anti-symmetric wavefunction, when two particles are interchanged.] Although first rank spinors are a non-faithful representation of the rotation group (e.g. electrons) as can be shown from studying unimodular group space, higher rank spinors can produce faithful representations e.g. photons. Pauli and Schwinger showed using the CPT theorem (the invariance of physical laws under reversal of charge, parity and time), half integer spinors have operators which are anti-symmetric i.e. fermions, while integer spinors operators are bosonic. This can be achieved directly from the group theory of higher rank spinors
Although not realising it at the time, Dirac was finding an equation that allowed compatibility between the unitary group of QM and the unimodular group of SR and it is this which produces g = 2, as well as invoking of anti-matter and the necessity of a field theory. [Unfortunately as Cartan has shown, such a spinor equation cannot be extended to the Riemannian techniques required by GR.] Historically, Dirac was motivated by the inadequacy of the Klein-Gordon equation. His main dissatisfaction was not that this relativistic version of QT could yield embarrassing negative values to the probability density, but rather that it did not conform to his beloved transformation theory, which he once remarked was his greatest single achievement. His contact transformations showed that QT could be expressed using a variety of bases e.g. momentum space. Just as GR was like a marble statue, which could be covered in an endless variety of covariant space-time coordinates, so could the tenets of QT be written in a countless choice of bases in Hilbert space.
The interchange of two non-distinguishable particles corresponds to 360 degrees of rotation in real space, which therefore corresponds to 180 degree of rotation of a (simple) spinor wavefunction. This does not affect the probability of a wavefunction being made, since this depends on square of the amplitude of the function, but it does change the sign of the wavefunction, which means that it must be anti-symmetric. Such anti-symmetric wavefuctions implies that no two quanta can occupy the same state i.e. Pauli’s exclusion principle is invoked!
What about the symmetric function of bosons? Well they are also unitary and unimodular, but the representation of the rotational group takes place on a space, which is the product of two spinors, which faithfully represent the rotational group. There is a great deal of group theory proving all this and it can be shown that an odd number of spinors multiplied together gives a non- faithful (fermionic) representation of the rotation group, while an even number gives a faithful representation (bosons). Incidentally, whereas a fermionic wavefunction undergoes a minus one phase change when 2 quanta are exchanged and bosons remain the same (plus one phase change) there are 2 dimensional systems called Anyons which can undergo a fractional complex phase change i.e. Any phase change angle, (hence the name) depending on the types of particles. [Such exotic quantum systems can be simulated, for example, when an electron gas is trapped in a 2D layer between two semiconductors.] The paths produced when 2 anyons are interchanged are topologically distinct in 2D, depending on the way the paths are braided These anyons may be useful in quantum computing, since the invariance of clockwise or anti-clockwise braiding can be used to overcome the loss of coherence which could leads to computer error.
To recapitulate, only suitable way of representing QM, (whose wavefunction evolves/transforms according to a unitary group) such that rotations of the X,Y and Z axes, can be catered for, is to resort to spinor space. The basic spinor rotates at half the rate as that of an Euclidean vector and hence it has an angular momentum of ½h and g = 2. This also causes an interchange of particles (= 360 degrees of rotation) to give a sign change, hence the function is antisymmetric, implying that no two particles can occupy the same state (fermionic statistics). For bosons, the spinor space is made up of an even number of spinors multiplied together (e.g. 2 for photons = spin 1h) and they faithfully represent the rotation group and have symmetrical wavefunctions. Indeed an even number of fermions may itself interacts with another of its ilk, in a bosonic fashion.
The more important group of transformations is the Lorentz group of special relativity (involving boosts in velocity as well as rotations). There is also a representation of this larger group in spinor space but although it is unimodular it is not unitary and the spinor has 4 components described by the transformation SU(2) * SU(2), (i.e. a pair of 2 spinors, each 2-spinor being related by parity). Moreover SR forms a non-compact group, while unitary transformations are compact and therefore SR is not compatible with basic QT. There is a theorem that states that there is no finite dimensional unitary representation of a non-compact Lie group and this overtly prevents a unitary group from representing the full Lorentz group. This leads to quantum field theory, which gets around the problem by introducing "functionals"—spin functions with an infinite degree of freedom. [For example Dirac’s equation in which there is a ‘four spinor’ composed of a pair of ‘two spinors’ related by parity.] Supernumbers also allows a way around this problem e.g. superstring theory.
Classically a particle has 3 degrees of freedom (X,Y & Z) while a field has infinite degrees of freedom each position has a field value). Quantum mechanically a particle (quanta) is represented by the wave function which also has 3 degrees of freedom W(x,y,z). There is a position operator and a momentum operator, which extracts eigenvalues when acting on the wavefunction and these correspond to the classical observables. For a (relativistic) field W(x,y,z) represents the amplitude of probability of finding a given distribution of particles/antiparticles of various energies and spin at that point (and takes the role of the position operator in that of a particle Lagrangian). Whereas Quantum mechanics allows the duality of particles and waves, quantum field theories allow a duality between matter and energy, string theory allows a unification of fermionic and bosonic fields. Remember that heisenberg's Uncertainty Principle relates to the complementary uncertainties in a particles position and momentum (~ rate of change of positioned) and when applied to fields, it implies that the more we know about the value of a field (number of particles at a location),.the less we know about its rate of change at that location.
A spinor field transform unimodularly and this causes the wavefunction (which has infinite degrees of freedom) to transform in the desired unitary way. [The field, being a ‘functional’ behaves as if it is a function with an infinite number of variables. It therefore avoids the problem of there being no finite unitary representation of a non compact group, such as the Lorentzgroup). This ensures that observable measurements are in agreement with special relativity as well as QM. In Dirac’s equation, no two spinors can occupy the same energy state (i.e. fermi statistics), while for photons, the likelihood increases and their (bosonic) statistics are expressed by the commutative relationships of the field operators.
As already mentioned, it was the quest to find a relativistic wave equation, which forced Dirac to introduce spinors as a necessity .[Cartan and later van derWaerden developed a complete theory of spinor calculus. The discovery of spinors came as something of a shock to the physics community, who up until then had thought that tensors were the only possible representation of the rotation group, while in actual fact if we relax the requirement of being faithful, we can allow the two-valued representation of spinors ]. This formulation also showed that particles had an inherent spin of 1/2h (electrons become inverted when rotated by 360 degrees), and the existence of anti-particles. [The prediction of anti particles, however is an abandonment of the Dirac equation as a single particle equation, since it is now required to describe both particles and antiparticles and we must therefore view it as a field equation] The only way that a spinor (which must obey a general unimodular transformation under a Lorentz transformation), can conform to the unitary transformation demanded by a quantum wave function, is if we move to what is sometimes referred to as second quantisation, in which a many particle wave function is achieved by changing the corresponding one particle wavefunction into a set of operators satisfying certain commutational relations. This results in a quantum field theory in which there is an indefinite number of particles distributed throughout space i.e;
QM + SR = Quantum field theory
In Dirac's equation we have moved from a function to the concept of a functional, which is a function with an infinite number of variables. As in the concept of a field, every position in space and time is given a variable value i.e we have infinite degrees of freedom. By considering functionals of spin functions and assuming that that the Hilbert space is a space of functionals, the spin can be extended so as to be in agreement not only with the group of rotations but also with the full Lorentz group! Under such a formalisation, when a Lorentz transformation is applied all the spin functions will undergo a unimodular transformation and hence behave like spinors, while spin functionals (the wave function psi) will undergo a unitary transformation, as required in order to preserve probability amplitudes. Dirac used his relativistic equation to obtain a method of describing the interaction of electrons with their electromagnetic field. This tentative way of doing electrodynamic calculations involved incorporating into the Dirac action, the Maxwell field, which although relativistic was not quantized (as was his electron field). This approach therefore considered the Dirac electron interacting with a classical Coulomb potential and although giving correct magnetic moment for the electron (Lande factor g=2), does not account for more subtle refinements such as the Lamb shift. There was already a way of expressing an electromagnetic field as an assembly of bosons, by utilizing Fock space, constructed from harmonic oscillators (together with the correct commutational relations for its creation and annihilation operators). However this was not a relativistic description of an assembly of photons. Later methods developed by Jordan, Heisenberg and Wigner etc., involving the canonical approach, did however produce a relativistic quantum field theory of electrodynamics. Alternatively, there is Feynman's path integral method which relies on a concept that was also initiated by Dirac. He elaborated it so as to incorporate interactions with quanta of electromagnetic fields (photons) as well as virtual particles and antiparticles of its own electron field. Quantum electrodynamics [QED], does however suffer from embarrassing infinities which occur when probing interactions at higher energies and these have to be removed by a process known as renormalization (a technique that does not however work for quantum gravity). Electrons however, are not the only source of electric charge and a more complete description would allow for quarks! Such Quantum Flavour and Colour Dynamics [QCD &QFD] that are incorporated in the Standard Model, rely on gauge theories in which there is spontaneous symmetry breaking that is crucial in explaining the mass of the gauge particles that carry the interaction (and indeed possibly explain the fermionic masses in Grand Unified Theories)
When a classical field is quantised, the quanta of the field are the particles represented by the classical field equations. Hence the relationship between the classical electromagnetic field and the photon (the descrete energy quanta of the radiation field) thus provides a a new insight into the interpretation of particles. The concept of such particles thus changes from that of the quntum mechanical 'wave-particle duality', to that of 'the quanta of a quantized field'. The electromagnetic field is however naturally relativistic (Lorenz covariant), so we need to find classical fields whose quanta correspond to 'matter' particles satisfying the Dirac or Klein Gorden equations. It is in this realization that we find one the fundamental conceptual shifts needed to proceed to the level of quantized fields. It is that relativistic quantum mechanical wave equations such as the Klein -Gordon and Dirac equations are to be reinterpreted as classical field equations at the same level as Maxwell's equation for the classical electromagnetic field. In general, in order to quantize a classical 'matter' field, we first express it in a Lagrangian form and find its corresponding (position and momentum) canonical conjugate variables -- the first of which I will denote as psi(x,t). [The required classical equation for the field is then usually retrieved by using Hamilton's equations]. Then we quantize the field by subjecting the canonical field variables to the correct Heisenberg's commutational relationships. The equation of motion for any quantum variable 'F' can then be obtained by replacing the Poisson bracket by the corresponding Heisenberg commutator bracket (which can be found since F and the Hamiltonian are given in terms of the known canonical variables). The quantum field quantity psi(x,t) is then regarded as an Hermitian operator (rather than a real numerical function) whose Fourier expansion can be written in terms of its creation and anihilation operators and in a relativistic field, we have to cater for both matter and ant-matter. These two operators must also satisfy commutational relationships, depending on whether the field represents an assembly of bosons or fermions.The quantity psi(x,t) plays a role in field theory analagous to that played by x, the position vector in particle (quantum) mechanics and the two canonical variables obey the same commutational rules of Heisenberg. The process of regarding ps(x,t)i as an operator rather than a number, is therefore part and parcel of the process of second quantization (i.e. the quantization of a field.) This field quantization has an obvious interpretation as a many particle theory, where the square of the amplitude is proportional to the number of particles present. An alternatively method of providing a quantum field theory, is to apply the path integral method and use Feynman propagators ( which are the inverse of the operator appearing in the quadratic part of the Lagrangian), to determine the contribution of each of the scattering probabilities of the particles interacting with their field.
Supersymmetry involves combining fermions (the basic constituents of matter) and bosons (responsible for the fundamental forces) on an equal footing in the same quantum field theory. As well as unifing fermions with bosons, it also unifies spacetime symmetries with internal symmetries, and (in the case of local supersymmetry), gravity with matter. The non compact groups of SR do not have a unitary representation in commuting numbers (e.g complex or real space) but there is a possibility in superspace. Bosons arise naturally as a field which provides the interaction between fermion particles, which usually manifest as a force (electromagnetism, strong nuclear interaction and gravity). Different fermions (e.g. electrons, neutrons) can be represented as an internal state, in which these components (fermions) are symmetric under certain local gauge transformations. [This state behaves like spinors and the gauges like SU(2)]. However the boson field associated with gravity (gravitons of spin 2h), must belong to the none compact group of SR, since GR which describes gravity, is a localised (hence generalised) form of SR. As mentioned above, we therefore need to resort to anti-commuting numbers in order to achieve this master supersymmetry and obtain unification. Fermions and bosons are therefore united in the same Quantum field theory and can be interchanged into each others state by symmetrical internal transformations e.g. electrons-sleptons or gravitons- gravitinos. Supersymmetry is an extension of ordinary Poincare space-time symmetry which is obtained by adjoining N spinorial generators whose anticommutator yields a translation generator (hence producing gravity when localized). This symmetry can be realized on ordinary fields (functions of space-time) by transformations that mix bosons and fermions but a more compact alternative to this component field approach is given by the superspace- superfield approach. Here superspace is an extension of ordinary space-time so as to include extra anticommuting coordinates in the form of N two component Weyl spinors $. Superfields Psi(£$) are then functions defined over this space. The transformations mixing bosons and fermions are then constant translations of the $ coordinate and also related rotations of the $ into the space-time coordinate £
In (closed) superstring theory, upon quantization, the canonical conjugate operators decompose into a Fourier series, which contain left and right handed harmonic oscillators which do not interact (that is as the string propagates, it has distinct right and left moving oscillator modes). Quanta are therefore represented by the dynamics of strings and different vibrational modes in different bosonic/ fermionic coordinates represent different quanta. The left moving modes are purely bosonic and exist in a 26 dimensional space, which has been compactified to 10 dimensions. The right moving modes only live in this 10 dimensional space and contain the supersymmetric part of the theory, this requires the introduction of superpartners that differ by a spin half (e.g. electron-slepton, photon-photino, etc) as well as the graviton/gravitino. The compactified 16 dimensional string lives on the root lattice space (e.g. 16D Tori), of an E8*E8 isospin (internal) symmetry, which is more than large enough to contain the required spectum of particles. When the left moving half and the right moving half are put together they produce the heterotic string (meaning "hybrid vigour"). Compactification of the extra six dimensions on a Ricci flat (e.g Calabi-Yau) manifold, then reduces the 10 dimensional superstring into our familiar 4 dimensional space-time, breaking the E8*E8 symmetry to produce the electroweak and strong interactions that are represented by U(1), SU(2) and SU(3) gauge symmetries. [ Under compactification that produces the Calabi-Yau space, one of the E8 groups is broken down to its E6 subgroup and E6 is a good candidate for a Grand Unified Theory, as it contains the subgroups of the Standard Model. The other E8 sector would describe (shadow) matter that could (only) interact gravitationally with the 'E6 matter']
Hence although the reason that fermions and bosons exist in subtle, deep and complex, if these symmetries were not enforced upon nature, we would not have the richness that we observe in the universe. This is nowhere more evident than in the creation of our own planet Earth and in the very laws that are responsible for its existence.
Regarding superunification it is not possible to satisfactorily incorporate gravity (which is governed by the non-compact Poincare group) in what is known as a Unitary representation (that dictates the other 3 quantum interactions), unless one resorts to supernumbers, which combines both fermions and bosons via supersymmetry. This concept was originally invoked in the early study of string theory but although this has received a recent surge in popularity, supersymmetry itself does not require a string formulation. I'll let Dirac have the last (controversial) word "Physical laws should have mathematical beauty" (epitaph)
