Lorentz Invariance and Second Quantization By treating electromagnetic modes in a cavity as a simple harmonic oscillator, with the oscillator level corresponding to the number of photons in the system of a particular energy, we were able to derive relations between the processes of stimulated and spontaneous emission and absorption from first principles. Dirac performed this calculation without recourse to statistical mechanics, relying instead only on the properties of the quantized wave equation for photons. It should not be surprising that the SHO algebra shows up, since each mode of the classical wave-in-a-box is itself a simple harmonic oscillator. We will now embark on what is referred to as second quantization, a phrase used to describe the canonical quantization of relativistic fields. 9 From now on, we will be setting both c = 1 and h = 1. Again, these factors can be restored in the end using dimensional analysis. It is conventional in this case to express all quantities in terms of energy. I.e. [length] = [time] Energy 1. [Momentum] = [mass] Energy. Lorentz Invariance 9 You may also be aware of the path integral formulation, where many things are more concise and easy to understand, although the connection between most physical observables is more opaque at times in this formulation. We will begin with second quantization (which is anyway necessary to make the connection between scattering amplitudes and the path integral formulation). Special relativity entails the addition of new symmetries of space and time. In non-relativistic systems, our symmetries were comprised of just rotations and translations in space and time. Special relativity requires that we enforce new symmetries on all our physical systems - results must also be invariant under boosts. 10 10 Usually more symmetry means easier to solve, however processes in special Rotations relativity can become very complex as final states can involve large numbers of particles when the energies associated Recall under two dimensional rotations, we have with particle scattering become large relative to rest mass energies - for x!x cos q + y sin q photons this is trivial since they always y! x sin q + y cos q (34) move at the speed of light, which is why radiation was the first subject of study for the pioneers of quantum which can be written in matrix form as theory. x y!! cos q sin q sin q cos q! x y! (35)
24 quantum field theory notes (hubisz - spring 2016) You can also write this in the more compact notation x i! R(q) j i x j (36) with i and j running from 1 to 2, and with an implied summation over the j index. Note we can also use the transposed rotation matrix to rotate row-vectors (which we denote with raised indices): x i! x j (R T (q)) i j = R(q) i j xj (37) Note that R 1 = R T, so that R T R = I, or R T (q) i k R j k = d j i This property of the rotation matrices (referred to as orthogonality) along with the trait that R preserves orientation characterizes the matrices R as rotations. This symmetry group goes by the name special orthogonal group of two dimensions or SO(2) for short. The group of 3D rotations is SO(3) (and so-on). The norm of vectors is invariant under rotations: x i x i! x j (R T ) i j R k i x k = x j d k j x k = x k x k, (38) and in fact this is another way to define the special orthogonal groups - the symmetries that preserve Euclidean norms x i x i = x i d ij x j. Lorentz Transformations Lorentz transformations are much like rotations, except that instead of leaving Euclidean norms invariant, they instead leave Minkowski norms invariant. That is, instead of preserving r 2 = x 2 + y 2 + z 2, they preserve s 2 = dt 2 dx 2 dy 2 dz 2. We use 4-vectors x µ =(t, x, y, z). We typically utilize Greek indices for 4-vectors (and higher rank objects). x 0 is taken to be the time component of a 4-vector. In order to preserve the norm, it is equivalent to state that Lorentz transformations leave the metric invariant: 0 1 1 L T 1 gl = g = B C (39) @ 1 A 1 This constraint on L is analogous to the relationship on Euclidean rotation matrices: R T R = R T IR = I. In flat Minkowski space, we will often interchange the metric g µn with h µn. Just as rotations leave Euclidean norms invariant due to the trigonometric relation sin 2 q + cos 2 b = 1, boosts leave Minkowski norms invariant due to a relation of hyperbolic trig functions cosh 2 b sinh 2 b = 1. One can identify the hyperbolic trig functions in terms of the boost velocities of the transformation: 11 11 b is sometimes referred to as the rapidity of the boost.
lorentz invariance and second quantization 25 cosh b = 1 p, and sinh b = v p 1 v 2 1 v 2 (40) In constructing a Lorentz invariant theory, we mean that the equations of the theory are the same irrespective of our inertial reference frame. We should also include invariance under translations in spacetime. The total group of space-time symmetries we would like to include are general Poincaré transformations. A Lorentz transformation acting on a space-time 4-vector can be represented by x µ = L µ nx n (41) where repeated Lorentz indices are summed (this is always the case, unless explicitly stated otherwise). L µ n is constrained to transformations which preserve the Minkowski distance between x µ and the origin, x 2 x µ x µ = g µn x µ x n = ~x 2 c 2 t 2. (42) This is what we mean by the Lorentz transformations being a symmetry. We did something to something, and it staid the same. The metric we showed above can be used to lower" indices, and the inverse metric, g µn can be used to raise" them. In order for the Lorentz transformation to preserve space-time vector distances, we require that g µn L µ rl n s = g rs. (43) Note that these transformations include ordinary spacial rotations as a sub-group, and that the entire set of Lorentz transformations also forms a group (it is not difficult to construct the inverse of L µ n using the metric tensor: (L 1 ) r n = Ln r ). We can write an infinitesimal Lorentz transformation as L µ n = d µ n + dw µ n (44) Multiplying this Lorentz transformation by its inverse constructed with g µn, we find that the infinitesimals must be anti-symmetric: dw µn = dw nµ. There are thus six infinitesimals, three of which span the usual spatial rotations, and 3 boosts, corresponding to shifts in inertial reference frame along any of the 3 spatial directions. However, not all Lorentz transformations can be obtained via repeated application of infinitesimal transformations. Taking the determinant of Eq. (43), we find that det L = ±1, while the determinant of Eq. (44) is always +1. Lorentz transformations with positive determinant are termed proper," and are connected in a continuous manner to the identity transformation L µ n = d µ n. Depending on the value of L 0 0, a Lorentz transformation is termed either orthochronous or non-orthochronous depending on whether
26 quantum field theory notes (hubisz - spring 2016) L 0 0 1 or L 0 0 apple 1 respectively (other values are forbidden, as can be derived from Eq. (43)). Orthochronous transformations form another subgroup of the Lorentz group. Clearly, all proper Lorentz transformations are also orthochronous. Discrete spacetime symmetries such as parity inversion and timereversal toggle between proper and non-proper Lorentz transformations, with time-reversal also toggling between orthochronous and non-orthochronous Lorentz transformations. Particle physics experiments have now shown us that parity and time-reversal are not fundamental symmetries, however, we still refer to the so called Standard model" of particle physics as being Lorentz invariant. Semantically, we refer to invariance under the proper orthochronous subgroup of the full Lorentz group when we say a theory is Lorentz invariant." Lorentz transformations on the Hilbert space - The Lorentz algebra So far we have only discussed the action of Lorentz transformations on the coordinates themselves. In the quantum theory, the action of Lorentz transformations on the Hilbert space is represented by unitary (or anti-unitary) operators which rotate the space of statevectors amongst each other. The operators which manifest this action must obey the usual composition rule, U(L 0 L)=U(L 0 )U(L) (45) For transformations which are close to the identity (and are proper / orthochronous), as in Eq. (44), we can Taylor expand the operator U in terms of the w s. We can write the result as U(L = 1 + dw) =I + i 2 h dw µnm µn. (46) There are six M s (they are anti-symmetric in the Lorentz indices), one for each w (M µn is also anti-symmetric in its indices). The six M s enact the rotations and boosts on the state-vectors in the Hilbert space. Remember that generators are the mathematical objects that span the possible directions" you can transform something in. They are basis vectors in the space of possible transformations. By repeated application of infinitesimal transformations, you can reach any transformation in the entire group (here, the proper orthochronous subgroup of the entire space of Lorentz transformations). Note that the generators carry Lorentz indices, just like the spacetime coordinates and the metric. Does this mean that Lorentz transformations mix the generators amongst each other, just as LT s mix
lorentz invariance and second quantization 27 time and space coordinates amongst each other? Yes! U 1 (L)M µn U(L) =L µ rl n sm rs (47) This is probably most obvious when you consider a Lorentz transformation acting on the momentum operator. The momentum operator, when acting on an eigenstate, has eigenvalue equal to the 4-momentum: P µ ~pi = p µ ~pi (48) In 4-vector notation, P 0 = H, and the spatial components of P µ are the usual 3-momentum operator. Since p µ should rotate by a usual Lorentz transformation, so should the operator P µ which produces this eigenvalue: U 1 (L)P µ U(L) =L µ np n (49) You can derive the commutation relations for the operators M µn using the constraints for Lorentz transformations. They are: [M µn, M rs ] = i h {g µr M ns g nr M µs g µs M nr + g ns M µr } (50) The 3 boost generators are the M 0i components of M µn (latin indices run over the spatial components i = 1, 2, 3, while greek indices run over all space-time components). The 3 generators which span the sub-group of spatial rotations are the M ij. Plugging boosts and rotations in different ways into Eq. (50), you find that The commutator of two rotation generators is another rotation generator (they form a subgroup!) The commutator of two boost generators is a rotation generator (a series of boosts can rotate your orientation) The commutator of a rotation generator with a boost generator is another boost generator (rotation changes the boost direction) We can also apply the rules for the transformation of the 4- momentum operator P µ in Eq. (49) to find the commutation relations between Lorentz generators and the Hamitonian and 3-momentum operator. rotation generators commute with P 0 = H (rotations leave the energy alone!) commutators of rotations with P i give a new P 0i (rotations rotate the direction of 3-momenta) commutators of boosts with H are non-zero (boosts change momentum!)
28 quantum field theory notes (hubisz - spring 2016) commutators of boosts with 3 momenta are proportional to H Scalar fields are functions of position and time that do not directly transform under Lorentz transformations. A Lorentz transformation acting on a scalar field acts only on the argument of the field: f(x)! f((l 1 ) µ nx n ) (51) The appearance of the inverse Lorentz transform corresponds to the fact that as you rotate your reference frame (for example) the field configuration appears to rotate in the opposite direction. Often this part of the transformation is irrelevant, and we speak of scalar fields as objects that do not transform under the Lorentz group.
Second Quantization Classical Field Equations Recall from our discussion on calculating the relationships between the Einstein coefficients that it was a quantization of a single mode of the electromagnetic wave spectrum that was key in our derivation. Of course we must quantize not just a single mode, but all of them in order to understand the various phenomena that can occur over a range of frequencies (i.e. processes with a possible continuum of final states as one finds in Compton scattering). Let us return to the classical wave equation, for which we have: f = µ µ f = 2 t ~r 2 f = 0 (52) The general set of classical solutions can be expressed as plane waves: f(x) =a p (t)e i~p ~x (53) or sines and cosines" where a p (t) satisfies the constraint 2 t + ~p 2 a p (t) =0, (54) which is the usual equation for a simple harmonic oscillator. If we were to generalize our earlier quantization of a single mode, which had a Hamiltonian given by H = w a wa w + 1, (55) 2 we would have for the finite box H = Â w ~k a ~ a k ~k + 1! L 3 2 ~ k d 3 p (2p) 3 w ~p a ~p a ~p + 1 2 (56) where the L! limit has been taken in the final equality. 12 We will return to this Hamiltonian later, and for now resume our discussion of the classical wave equation. A generic real solution to these equations can be written as f(x, t) = d 3 p h (2p) 3 a p (t)e i~p ~x + a p(t)e i~p ~xi (57) 12 Due to the volume factor in the integral, we often refer instead to a Hamiltonian density operator H.
30 quantum field theory notes (hubisz - spring 2016) Since the SHO solutions for a p are also exponentials, we can then write the full solution as f(x, t) = d 3 p h (2p) 3 a p e ipx + a pe ipxi (58) where now a p are just numbers 13 and the products px in the exponentials are 4-vector dot products px = p µ x µ. In addition, we have the constraint that p 0 = w p = ~p. This equation of motion is referred to as the massless Klein- Gordon field equation. In an infinite sized box, there are modes with all frequencies accessible - there is no gap" in the spectrum of classical states. The same sort of equation appears in classical electromagnetism. In Lorenz gauge, where µ A µ = 0, Maxwell s laws are (in the absence of sources) 13 In principle, there are other terms possible here, but you can use change of variables ~p! ~p to reduce it to this form. µ F µn = A n n ( µ A µ )= A n = 0 (59) meaning that every component of the 4-vector potential obeys a classical equation identical to the massless Klein-Gordon equation, and thus has a space of solutions identical to Eq. (58) Hamiltonian for massless fields For our quantized massless fields, we then have the following Hamiltonian: H = L 3 d 3 p (2p) 3 w ~p a ~p a ~p + 1. (60) 2 Using the ladder operators in the context of quantized classical fields is referred to as second quantization 14 In second quantization, the new features relative to single particle quantum mechanics are as follows: (1) We have an infinite number of single particle" systems - one for each ~p. 14 This arises due to the classical discretuum of solutions in a finite volume, and their subsequent quantization, so it is not really a second quantization of anything. It is merely a single quantization each of an infinite number of classical SHO s. (2) We have an interpretation of the energy levels for a single ~p oscillator as the number of particles in that particular mode. Note that our Hilbert space is now quite large. It is a Fock space - the direct sum over Hilbert spaces associated with n-particle states: F = n H n (61) For example, for one E particle type, a generic state in H n can be written as p µ 1,, pµ n where the momenta each satisfy p 2 i = m 2. 15 15 It is important that the Fock space contains only physical states - i.e. those satisfying the relevant energymomentum conditions for particles.
second quantization 31 Field Expansion We would like to understand how to obtain the Hamiltonian in Eq. (60). In taking the infinite volume limit, our expressions for the commutation relations between the annihilation and creation operators takes a different form. The correct generalization is h i i a ~k, a ~ k 0 = d ~k, ~ k 0 $ ha ~p, a ~p 0 = (2p) 3 d 3 ~p ~p 0 (62) The factors of 2p are part of a convention we chose as we took the infinite volume limit of the finite volume Hamiltonian. These creation operators create particles with momentum ~p: a ~p 0i = 1 p 2wp ~pi (63) The factor of 1/ p 2w p is another convention that will make our lives easier when we write down our fields later on. Note that with this choice of the spectrum of states created by the a s, we have the following normalizations (given h0 0i = 1): ~p ~p 0 = 2 p w ~p w ~p 0 D0 a ~p a ~p 0 0 E = 2w ~p (2p) 3 d 3 ~p ~p 0 (64) Thus by summing over a complete set of states, we see that the identity operator on the subspace of single particle states is d 1 1-particle = 3 p 1 (2p) 3 ~pih~p (65) 2w ~p Now recall our classical massless Klein-Gordan equation. We wrote the solution in terms of c-number coefficients a p, but of course we could have made those coefficients whatever kind of mathematical objects we liked. We define a quantum field that satisfies this equation as 16 f 0 (x) = d 3 p 1 h (2p) 3 a 2w ~p e ipx + a ~p eipxi (66) p Note that we have written f 0 as an operator which is explicitly a function of time. 17 This is the case in the Heisenberg picture, where states in the Hilbert space carry no time dependence, and where instead it is the operators acting on the Hilbert space that contain this dependence. 18 So since f 0 is an operator, it is worth finding out what it does. Let us consider the projection of the state that f 0 creates from the vacuum onto a single-particle state of momentum ~p: * + d h~p f 0 (x) 0i = 0 q2w 3 k 1 p a p (2p) 3 p a k e ikx + a k eikx 0 2wk = d 3 r k wp h (2p) 3 e ikx 0 a p a w k 0 D Ei + e ikx 0 a p a k 0 k =e ikx (67) 16 This is just a definition - there are many possible choices of operators that satisfy the classical wave equation, but this is going to be a definition which is of particular value to us. 17 The 0 subscript will denote that we are talking about free field without interactions. When we consider perturbations that we associate with particle interactions we will drop this subscript. 18 This is of utility in our relativistic theory, since a separation in the treatment of time and spatial coordinates would make the symmetries of Poincaré invariance hard to identify. In the Heisenberg picture, it is beneficial that both time and space coordinates are on equal footing - as labels on operators.
32 quantum field theory notes (hubisz - spring 2016) Note that this is a relativistically invariant version of the familiar relation h~p ~xi = e i ~ k ~x. So what f creates is a particle localized at position x. 19 As a check that this field we have defined is consistent with our expression for the Hamiltonian for classical massless fields, we evaluate the commutator of f 0 with H 0. We should see that it is consistent with the operator evolution equation in the Heisenberg formalism: [H 0, f 0 (x)] = = d 3 p (2p) 3 d 3 p 1 h (2p) 3 p 2wp d 3 apple k 1 (2p) 3 p w p a 2wk pa p + 1, a 2 k e ikx + a e ikx w p a p e ipx + w p a pe ipxi = i t f 0 (x) (68) 19 This is not surprising. If you want to create a particle at position x, it is analogous (classically) to exciting a delta function spike along a string. For a finite string, the Fourier coefficients are each B n µ e iknx, where k n is the wavenumber of the particular mode, so that d(x) Â n e iknx Y n (x 0 ) where Y n are the eigenfunctions of the wave equation. In the continuum limit, this becomes essentially identical in form to our quantum field, where the annihilation and creation operators stand in for the eigenfunctions. It is often useful to use the same notation for interacting fields as we use for free fields: f(~x, t) = d 3 p 1 (2p) 3 p ha p (t)e ipx + a 2wp p(t)e ipxi (69) where a and a satisfy the usual algbra at any fixed time, so they always create states with momentum ~p. However, if f is to be a solution of the equation of motion that includes interaction (nonlinear) terms, then there will be non-trivial evolution of a p and a p. 20 The interpretation of this is that the time evolved annihilation and creation operators do not necessarily create single particle states at all times t. 20 This evolution will be slow if the interaction terms are small