November 2016 PX430: Gauge Theories for Particle Physics Tim Gershon (T.J.Gershon@warwick.ac.uk) Handout 1: Revision, Notation and The Gauge Principle Relativistic quantum mechanics is a prerequisite for this module, and therefore we will assume knowledge of the material covered in PX408, available from http://www2.warwick.ac.uk/fac/sci/physics/teach/module home/px408/. Nonetheless, let s start with some very brief revision. Relativity Einstein s principle of special relativity can be formulated thus The laws of physics are the same in all inertial frames of reference. This has several far reaching consequences, including the fact that space and time, rather than being distinct entities, become mixed together as components of the 4-dimensional space-time any relativistically invariant theory has to treat space and time on the same footing. This can be made manifest using four-vector notation, since the product of a covariant 4-vector and a contravariant 4-vector is invariant. Quantum Mechanics Most of the important consequences of quantum physics will be relevant to this module, including the concept of spin, Heisenberg s uncertainty principle (especially when stated in the form that the commutator of position and momentum operators gives in natural units the imaginary number: [ˆx, ˆp] = i), Heisenberg s equation of motion ( Â ] = i [Â, Ĥ, so that any operator that commutes with the Hamiltonian represents a constant of motion), the Pauli exclusion principle, and so on. For our purposes one particularly important outcome of quantum theory is the wavefunction interpretation: the state of a system is described by a complex valued wavefunction (let s call it ψ), with the square of the magnitude of the wavefunction ( ψ 2 ) giving the probability density (this is sometimes called the Born interpretation). Relativistic Quantum Mechanics We have previously encountered two relativistic wave equations: the Klein-Gordon equation and the Dirac equation ( µ µ + m 2 )φ = 0, (1) (iγ µ µ m)ψ = 0, (2) where the gamma matrices are defined by their anticommutation relations {γ µ, γ ν } = 2g µν. We have seen that the solutions to the Dirac equation are four-component spinors, and that consequently the quantum property of spin and the existence of antiparticles emerge as natural outcomes of the theory. We have also discussed the limitations of relativistic quantum mechanics, specifically the need for a theory that naturally handles particle-antiparticle pair creation and annihilation, and the experimental evidence that such a theory is necessary.
Hamiltonian & Lagrangian Mechanics It can be difficult to avoid getting Hamiltonian & Lagrangian mechanics mixed up, since, for example, the starting point for Lagrangian mechanics is Hamilton s principle of least action. Without worrying too much about which is which, we define the action S = Ldt (3) where L is the Lagrangian, and the integral is between initial and final points. In classical physics, the Lagrangian is usually defined as the difference between the kinetic and potential energies: L = T V. Considering a particle travelling between initial and final points, the principle of least action states that the path taken will be the one that minimises the action. Writing the Lagrangian L = L(q i, q i ) in terms of generalised coordinates q i (t) and their derivatives with respect to time q i (t), allows us to obtain the Euler-Lagrange equations. The derivation goes as follows: S = δs = 0 = 0 = 0 = 0 = L(q i, q i )dt, (4) δl(q i, q i )dt, (5) ( ( ) ( ) ) L L δq i + δ q i dt, (6) q i i q i ( ( ) ( ) ) L dqi L d q i i dt + qi dt, (7) q i dt ( ( ) L q i d ( ) ) L q i dt, (8) q i dt q i i where in the last stage we have used integration by parts, uv = [uv] B u v, and used boundary conditions to identify that the constant term goes to zero. Since each term of q i is, in general, non-zero, this identity is only satisfied when: ( ) d L L = 0. (9) dt q i q i These are the Euler-Lagrange equations. Q1 Work carefully through the steps of the derivation of the Euler-Lagrange equations (Eq. (9). Q2 For a classical particle in a potential V = V (q i ), write down the kinetic energy T, and show that the solution of the Euler-Lagrange equations gives Newton s law, F = ma. Q3 Consider further a particle in an electromagnetic field, with kinetic energy given by T = 1 2m (p q)2. Solve the Euler-Lagrange equations for this case. It can be preferable to replace the generalised velocities q i by the corresponding conjugate momenta, defined as p i = L q i (10) Q4 For the classical Lagrangian discussed above, what are p i?
The Hamiltonian is then defined as (here including explicitly a possible dependence on t) H(q i, p i, t) = i q i p i L(q j, q j, t) (11) Q5 For the classical Lagrangian discussed above, what is H? Perhaps surprisingly, this Lagrangian-Hamiltonian formulation of mechanics, which appears initially to be little more than an interesting digression in classical physics, turns out to be extremely useful in quantum physics, and indeed in quantum field theories. In such cases, it is more common to consider the Lagrangian density, L, such that the action is obtained from the integral over all space and time dimensions S = L d 4 x (12) Writing the Lagrangian density as a function of a field ψ, which itself is a function of (generalised) coordinates x µ, the quantum field theoretic version of Eq. (9) is ( ) L µ L ( µ ψ) ψ = 0. (13) (ctually, in quantum field theory we should really replace both the field and the Lagrangian density with operators: ψ ˆψ, µ ψ µ ˆψ and L ˆL.) Q6 Write Eq. (13) in full, expanding the Lorentz invariant term into,, etc. Essence of the Gauge Principle s mentioned above, the wavefunction interpretation of QM tells us that physical systems are defined by a complex wavefunction. Such a complex wavefunction has two degrees of freedom, which we can write as the real and imaginary parts, or as the magnitude and the phase. Since all physically observable quantities depend only on the magnitude of the wavefunction (or rather, the magnitude squared, which gives the probability density), the phase is not an observable quantity. Consider a complex wavefunction ψ = ψ e i arg(ψ). Suppose this is made up of two interfering terms, then the observable quantity is ψ 2 = ψ 1 + ψ 2 2 = ψ 1 2 + ψ 2 2 + 2 Re(ψ 1 ψ 2) = ψ 1 2 + ψ 2 2 + 2 ψ 1 ψ 2 cos(arg(ψ 1 ) arg(ψ 2 )) (14) Clearly this depends on the relative phase between the two interfering components. There are very many examples of such effects, the most famous being Young s double slit experiment. So relative phases (ie. phase differences) are observable, but absolute phases are not. Consequently, there is a symmetry inherent in any quantum mechanical theory: if the wavefunction is changed by a (global) phase factor, the physics remains unaffected. Such a change of phase is known as a (global) gauge symmetry. Since Noether s theorem states that for every symmetry there is an associated conserved quantity, it is natural to ask what is the conserved quantity associated with the above global gauge symmetry? This is difficult to answer, since the question has been posed in very general terms. However, bearing in mind that we will be principally concerned with the electromagnetic interaction and associated wavefunctions, it might be possible to guess that the answer is the electric charge. Q7 Familiarise yourself with Noether s theorem. How does it apply to a) continuous symmetries and b) discrete symmetries?
Q8 What is the difference between a gauge symmetry, and, for example, the symmetry with respect to translation of spatial coordinates that is related to conservation of momentum? The gauge principle states that such symmetries of the Lagrangian with respect to rotations in some internal space are of profound importance. We shall be concerned not only with global gauge transformations, but also with local transformations, where the change of phase can vary in space and time. s we shall see, the requirement that our theory be invariant under such gauge transformations can be used to determine the entire dynamics of interacting (quantum) gauge fields. Gauge Invariance of Maxwell s Equations Before turning to quantum fields, we can get an idea of the principle from classical physics. We start with Maxwell s equations (in Heaviside-Lorentz units):.e = ρ em E = B.B = 0 B = j em + E together with the continuity equation: Gauss Faraday Lenz no magnetic monopoles mpère Maxwell (15) ρ em +.j em = 0. (16) We can write Eq. (15) in terms of the vector potential µ = (V, ), where B = and E = V. This is achieved by defining the field strength tensor from which Maxwell s equations can be written where j ν em = (ρ em, j em ). Q9 Show that Eq. (18) implies all of Eq. (15) as well as Eq. (16). F µν = µ ν ν µ, (17) Q10 Show that F µν is antisymmetric. What are the elements of F µν? We now consider transformations of the form µ F µν = j ν em, (18) µ µ = µ µ χ (19) where χ is an arbitrary real-valued scalar function, i.e. it depends on 4-position, but its value is a real number. It should be clear that this transformation of µ has no effect on physics, since both B and E are unaffected; this is a local gauge transformation. Q11 Satisfy yourself that both B and E are unaffected by the gauge transformation of Eq. (19). Q12 Show the the field strength tensor F µν is unaffected by the gauge transformation of Eq. (19).
The fact that we can change µ without affecting physics allows us to choose a particular form with which it is convenient to work. This is called choosing a gauge. One popular choice is called the Lorenz gauge, µ µ = 0. [n.b. Lorenz Lorentz!] With this choice, and after combining Eqs. 17 and 18 to get ν ν ( µ µ ) = j ν em (20) we find ν = j ν em. (21) From here we can see that each component of µ satisfies the massless Klein-Gordon equation in free space (j ν em = 0). Q13 Suppose that µ µ = f 0. Show that a gauge transformation can be applied to change to where µ µ = 0. What (in terms of f) is the form of the transformation? Q14 In fact, the Lorenz gauge condition does not specify µ uniquely. Show that further gauge transformations of the form µ µ = µ µ Λ, where Λ is any scalar function that satisfies Λ = 0 will (i) leave F µν (and hence the observable physics) unaffected, and (ii) preserve the Lorenz condition. Q15 The Coulomb gauge specifies. = 0. If the Coulomb gauge is satisfied, find an expression for the Lorenz condition. How many independent components does µ have? The fact that expressing Maxwell s equations in the manifestly gauge invariant form of Eq. (18), automatically embodies the continuity equation, suggests that there is a close relation between gauge invariance and charge conservation. Gauge Invariance in Quantum Mechanics Starting with the Schrödinger equation for a free particle Hψ = 1 2m ( i )2 ψ = i ψ we can introduce the electromagnetic interaction via the minimal substitution (22) D = iq, D0 = + iqv ( µ D µ = µ + iq µ ) (23) where the contravariant form is in parentheses since we do not expect to be able to apply a Lorentz invariant treatment to the non-relativistic Schrödinger equation. We then find the Schrödinger equation for a particle in an electromagnetic field ( ) 1 Hψ = 2m ( i q)2 + qv ψ = i ψ. (24) Q16 Show that the classical expression for the force on a charged particle in an electromagnetic field, F = qe + qv B, where v is the particle s velocity, can be derived from Eq. (24), using Lagrange-Hamilton mechanics. We now apply a gauge transformation. Both ψ and µ are transformed. t first, let us not specify the form of the transformation, but simply replace these variables by their primed versions: ( ) 1 H ψ = 2m ( i q ) 2 + qv ψ = i ψ. (25)
We require that physics should be unaffected under this transformation, which means that we need to obtain an equation of the same form (it does not have to be exactly the same equation, just to take the same form). If the transformation of µ has the form of Eq. (19), it is clear that we need to make some transformation ψ ψ in order to keep the physics unaffected. What is the form of this transformation? Since ψ 2 is an observable quantity, we require ψ 2 = ψ 2, and so we are only allowed to multiply by a phase factor. In fact, the answer is ψ ψ = e iqχ ψ. To see this, consider ( i q )ψ = ( i q q( χ))(e iqχ ψ) = e iqχ ( i q)ψ (26) and (i ) ( qv ψ = i ) χ qv + q (e iqχ ψ) ( = e iqχ i ) qv ψ (27) i.e. id ψ = e iqχ ( idψ) & id 0 ψ = e iqχ (id 0 ψ) (28) and so we find 1 2m ( id ) 2 ψ = e iqχ 1 2m ( id)2 ψ = e iqχ id 0 ψ = id 0 ψ. (29) We have recovered the Schrödinger equation for a particle in an electromagnetic field. Q17 Prove Eqs. (26) and (27). Q18 Prove Eq. (29). Therefore, the gauge invariance of Maxwell s equations remains an invariance in quantum mechanics, provided we make the combined transformation The Gauge Principle µ µ = µ µ χ & ψ ψ = e iqχ ψ. (30) What we have shown above is that if we make the gauge transformation µ µ = µ µ χ, then we require a transformation of the wavefunction of the form ψ ψ = e iqχ ψ in order to recover the same physics. The gauge principle, which is ultimately more powerful, makes the reverse argument: we require physics to be invariant under the local gauge transformation ψ ψ = e iqχ ψ, and consider the effect on derivative terms: µ ψ µ ψ = e iqχ µ ψ + iq ( µ χ) e iqχ ψ. (31) We can then see that D µ ψ = ( µ + iq µ ) ψ D µ ψ = e iqχ D µ ψ if µ µ = µ µ χ. Thus we can determine both the transformation required of µ and the form of the minimal substitution (and the reason for calling it the minimal substitution becomes clear to have local gauge invariance we need to introduce a field µ and this is the simplest way to do it). In other words starting from a free particle equation, by making a local gauge transformation, we can determine the form of µ and of the minimal substitution itself. Q19 Start from the free particle Schrödinger equation, and by requiring physics to be invariant under the local gauge transformation ψ ψ = e iqχ ψ, determine the form of the electromagnetic gauge field µ.
It might be reasonable at this stage to ask if there are any other gauge transformations we could make to our wavefunction ψ, which might determine interactions other than the electromagnetic one. We will address this question later. For now, we simply note that the transformation ψ ψ = e iqχ ψ is a phase change, and can be thought of as a rotation in the internal (real, imaginary) space of the wavefunction. This is called a U(1) gauge transformation. Successive transformations of this form can be applied without regard to the order i.e. U(1) rotations commute. This is called an belian gauge transformation we will encounter the non-belian (non-commuting) case later. [side: U(1) is a symmetry group, sometimes called the circle group. It is the multiplicative group of all complex numbers with unit magnitude. The mathematical language of group theory is quite helpful to discuss the more complicated gauge transformations that we will encounter at the end of the module. Unfortunately (or perhaps fortunately, depending on your perspective), time limitations prevent any proper discussion of this aspect. The interested student is encouraged to read further into this topic, for example ppendix M of Volume 2 of Gauge Theories in Particle Physics, by itchison & Hey, 3 rd edition, or Chapter 4 of Mathematical Methods for Physicists by rfken and Webber (6 th edition). There is also some material available on the web that is associated to relevant modules offered in the Mathematics department: M249 Groups and Rings, M3E1 Groups and Representations and M4E0 Lie Groups, though parts of these modules may not be at the appropriate level.] The Gauge Principle in Classical Electromagnetism s an application of the gauge principle, let s go back and consider electromagnetism. We know that we can form a theory around a Lagrangian density that is both (i) gauge invariant and (ii) Lorentz invariant. We already have an object which satisfies (i), namely F µν, and to satisfy (ii) we will need to contract the indices. So the only possibility is L = 1 4 F µνf µν, (32) where the factor of 1 4 is conventional, but the minus sign is not (remember, we need to minimise the action). So, simply requiring that the theory should be invariant under local gauge transformations, as well as Lorentz invariant, is enough to derive electromagnetism, as well as the form of the interaction between electrons and photons (electrodynamics). Clearly this is a very powerful principle. Gauge Invariance in Relativistic Quantum Mechanics Naturally, we would like to extend the above discussion to a relativistic quantum mechanical treatment, to see how the particles described by the Klein-Gordon and Dirac equations (i.e. the solutions to those equations) interact with the associated gauge fields. Indeed, there is nothing to stop us from trying to do so: we can either apply to minimal substitution µ D µ = µ + iq µ to introduce electromagnetic interactions, then gauge transform µ and see what happens, or alternatively we could attempt to apply the gauge principle, and assert that physics should be invariant under U(1) transformations. However, the effects are much clearer when we use the Lagrangian formulation within quantum field theory. Q20 pply the minimal substitution to (i) the free particle Klein-Gordon equation ( µ µ +m 2 )φ = 0, (ii) the free particle Dirac equation (iγ µ µ m)ψ = 0. Obtain conserved currents in both cases.