Why quantum field theory?

Transcription

1 Why quantum field theory? It is often said that quantum field theory is the natural marriage of Einstein s special theory of relativity and the quantum theory. The point of this section will be to motivate this statement. We will, first of all, put forward an attempt to convince you that there is a conflict between the quantum mechanical theory of a single particle and the special theory of relativity, that is, that the two are fundamentally incompatible. Then, we will discuss how quantum field theory can come to the rescue and resolve this conflict. The upshot is the statement that, if you want to use quantum mechanics and you want it to be relativistic, you need a generalization of single-particle quantum mechanics which contains states with different numbers of particles. For this, quantum field theory turns out to be an adequate solution to the problem. Before we get on with it, though, let us also state that, in the near future, we will see a construction that shows us that quantum field theory is also a natural framework in which we can study quantum mechanical systems where different quantum states can have different numbers of particles. In fact, this can be very useful even in non-relativistic physics where we are not forced to use it, or at least there is no basic principle which tells us we must use it, but where it turns out to give us a convenient approach to many physical problems. The idea that a significant generalization of single-particle quantum mechanics might be what is needed in order to describe particles in a way that is compatible with the special theory of relativity might already be apparent to the reader. A very important part of special relativity is the equivalence of energy and mass, as in Einstein s famous formula E = mc 2. According to this formula, as far as special relativity is concerned, a particle can be replaced by some energy. More concretely, we could consider a particle and an anti-particle. In relativistic physics, they can combine and annihilate, leaving behind a puff of energy, in the form of photons. To describe this physical process, we would need a framework wherein there is a quantum state with a particle and an anti-particle, for example, an electron and a positron, and another distinct quantum state where there is just the product of their annihilation, some photons. This is a first criterion. Secondly, we would also need to be able to analyze the transition between the two states,

2 for example, to predict the probability amplitude for this transition to occur. We will eventually see that a natural framework where both of these criteria are satisfied is quantum field theory. The above argument might already be enough motivation to study quantum field theory. However, here, we want to emphasize the point by asking what happens if we simply generalize what we know about single particle quantum mechanics to the situation where the particle can have very large energies, so that its total kinetic energy can have an order of magnitude that is comparable to its rest mass. As we shall see, we will encounter problems when we try to describe a single particle which obeys the rules of quantum mechanics and also obeys the rules of the special theory of relativity. Let us begin by considering a single particle traveling on open, infinite three dimensional space. This particle should have a conserved linear momentum, p, and a conserved total energy, E. According to special relativity, the energy of the particle and momentum of the particle are related to each other by the equation E( p) = m 2 c 4 + p 2 c 2 () where m is the rest mass of the particle and c is the speed of light. In the quantum mechanics of a single particle, we could consider a quantum state of the particle which is an eigenstate of its linear momentum, ˆp i p = p i p, i =, 2, 3 (2) where we have denoted the momentum operator by ˆp, with Cartesian components ˆp i, the eigenvalue of the momentum operator by p i and the eigenstate of the momentum operator with eigenvalue p i by the ket p. The states with definite momentum have a continuum normalization, so that the product of a bra and a ket is p p = δ 3 ( p p ). Because the energy of the particle is a simple function of the momentum momentum of the particle, given in equation () above, an eigenstate of the momentum is also an eigenstate of the energy, that is H p = m 2 c 4 + p 2 c 2 p (3) where H is the Hamiltonian. Here, we are assuming that there is a Hamiltonian operator whose eigenvalues are the energy states of the particle. Once 2

3 we have the Hamiltonian, we can form the Schrödinger equation which must be satisfied by the time-dependent state vector of the quantum mechanical particle. We will denote this state by Ψ(t) and the Schrödinger equation is i h Ψ(t) = H Ψ(t) (4) t The solution of this equation, assuming that at t = 0 the particle is in an eigenstate of momentum p is Ψ( p, t) = e i m 2 c 4 + p 2 c 2 t/ h p (5) for any value of the momentum vector p. This simple development would seem to be a complete solution of the quantum theory of a single relativistic particle. It allows us to ask any question about it. For example, let us consider the scenario where, at some initial time, say t = 0 the particle is localized at x = 0. This state could be created by a measurement of the position of the particle, for example, which we are assuming can have arbitrarily good resolution. We construct an eigenstate of position, that is, one which obeys ˆx i x = x i x (6) by superposing the complete set of momentum states as x = d 3 p p p x (7) where the overlap matrix is a plane wave p x = ei p x/ h (2π h) 3 2 (8) Then, the wave function that evolves from an eigenstate of position, at a time t later, is e iht/ h x = d 3 p p e i m 2 c 4 + p 2 c 2 t/ h+i p x/ h (9) (2π h) 3 2 We can now ask the question as to the amplitude for observing the particle at point y after a time t has elapsed. The answer is simply the overlap of 3

4 the position eigenstate y with the above wave function evaluated at t. The result is y e iht/ h x = d 3 p e i m 2 c 4 + p 2 c 2 t/ h+i p ( x y)/ h (0) (2π h) 3 Now, we find the difficulty. One of the postulates of the special theory of relativity states that the speed of light is a maximum speed. However, from equation (0), the probability amplitude is nonzero in the causally forbidden region, where y x > ct. There seems to be a nonzero amplitude for motion at speeds greater than that of light. A formal way to see that (0) is indeed nonzero in the forbidden region, is to consider t where it occurs in that equation as a complex variable. Then, (0) is analytic in the lower half of the complex t-plane. When t is real, the expression is a distribution which should be defined by its limit as complex t approaches the real axis from the lower half plane. Given that it is analytic in this domain, it cannot be zero in any region of the lower half plane plus the real axis except for discrete points, otherwise it would have to be zero everywhere. It is definitely not zero for all times, in fact when t = 0 it is a Dirac delta function. Thus, it cannot be zero in the entire region ct < y x. To see this more explicitly, we can do the integral for the special case where m = 0. It becomes y e iht/ h x = 4π 2 h 2 pdp [ e ip[ct x y ]/ h e ip[ct+ x y ]/ h] i x y 0 [ ] = lim i ɛ 0 + (ct) 4π 2 x y ct x y iɛ ct + x y iɛ = [ ] iπδ((ct) 2 x y 2 P )sign(t) + () (ct) 2π 2 (ct) 2 x y 2 In the second line above, we have defined the integral over the semi-infinite domain by introducing the positive infinitesimal parameter ɛ. This is tantamount to defining the distribution that we obtain as a limit as t approaches the real axis from the lower half of the complex plane. We have used the identity x iɛ = P x + iπδ(x) This is in addition to the already obvious difficulty that the expression (0) is not Lorentz invariant. In fact, it transforms like the time derivative of a Lorentz invariant function. Let us overlook this issue for the time being. 4

5 where P/x is the principal value distribution. Also, for the Dirac delta function δ(x 2 a 2 ) = (δ(x a) + δ(x + a)) 2 a In equation (), we see that the wave-function of a massless particle spreads in two ways. The first is a wave which travels at the speed of light and is therefore confined to the light cone - where x y = ct. The second is a principle value distribution which is non-zero everywhere, including in the forbidden region where x y > ct. This latter spreading of the wave packet violates causality. It tells us that, in our quantum mechanical system, the result of a measurement of the position of the particle at position y after time t could have a positive result. The particle could be observed as travelling faster than light. This would certainly seem to be incompatible with the principles of the special theory of relativity where objects are restricted to having subluminal speeds. Now that we have found a difficulty with causality, we need to find a way to resolve it. We will resolve it by going beyond single-particle quantum mechanics to an extended theory where there is another process which competes with the one that we have described. The total amplitude will then be the sum of the amplitudes for the two processes and we will rely on destructive interference of the amplitudes to solve our problem, that is, to make the probability of detecting the particle identically zero in the entire forbidden region x y > ct. To include the second process, we will first frame the first process, the one we have discussed so far, as the following thought experiment. One observer, whom we shall all Alice, is located at position x and prepares the particle in the state which is localized at x. Alice could do this by measuring the position of the particle and we assume that the result of the measurement is that the particle is at position x. We assume that Alice can do this measurement with arbitrarily good precision. Immediately after the measurement, the particle is allowed to evolve by its natural time evolution, the one which we have described above, so that after time t, its quantum state is given by equation (9) and its wave-function by equation (0). Then, at time t, another observer, Bob, who is located at point y does an experiment to detect the particle. Of course, in a given experiment, Bob might or might not find the particle at y. But, given that the particle is has non-zero amplitude to propagate there, if Alice and Bob repeat this experiment sufficiently many times, Bob will eventually detect the particle at y. The result of the experiment is to collapse 5

6 Figure : The wave packet is initially localized at x and as time evolves it spreads in such a way that there is a nonzero amplitude for detecting it in the vicinity of point y. If it is detected at y, since ct < x y, its classical velocity would be greater than that of light. the particle s wave function to one which is localized at y. The amplitude for the particle to propagate to y is given by (0). If this were all there is to it, the result of the experiment violates causality. The second process that we will superpose with the one that we have described will require other states to be introduced. It then clearly involves an extension of single particle quantum mechanics. In the second process, the attempt by Bob, the observer who is located at y, to observe an electron s position creates a pair consisting of a particle and an anti-particle. The position measure collapses the wave function of the particle into the position eigenstate localized at y, the position which was the final state of the particle in the first experiment. The anti-particle is interpreted as a particle which 6

7 Figure 2: We should add to the amplitude for the particle to travel from x to y as in figure the amplitude that a particle-anti-particle pair is created at y, the particle continues forward in time as it did in the first process, the anti-particle propagates backward in time and annihilates the particle which was prepared in the state localized at x. moves backward in time, from time t to time 0. 2 After time t it has an amplitude to arrive at position x where it annihilates the particle that Alice, the observer at x, has prepared in the localized state. The result of this second process is the same as that of the first process, a particle begins in a state localized at x and after a time t it is detected in a state localized at y. The amplitude for the second process is similar but not identical to that of the first process, due to the fact that the positron propagates backward in 2 The interpretation of the anti-particle as a particle which moves backward in time originates with Stueckelberg [] and it was adopted by Feynman [2] in his formulation of quantum electrodynamics. 7

8 time. It is x e iht/ h y antiparticle = d 3 p ei m 2 c 4 + p 2 c 2 t/ h i p ( x y)/ h (2π h) 3 (2) The total amplitude is the sum of amplitudes of the two processes, A = y e iht/ h x particle + y e iht/ h x antiparticle (3) = d 3 ei p ( x y)/ h [ ] p e i m 2 c 4 + p 2 c m 2t/ h + e i 2 c 4 + p 2 c 2 t/ h (4) (2π h) 3 Now, the total expression can have destructive interference. We will not demonstrate it in the general case, but in the case where the mass of the particle and antiparticle is zero. There, we can perform integral in (2) explicitly, x e iht/ h y antiparticle = [ ] iπδ((ct) 2 x y 2 P )sign(t) (ct) 2π 2 (ct) 2 x y 2 (5) We see that, like the amplitude for the particle, the amplitude for the antiparticle also spreads outside of its light cone. However, when we add the amplitudes of the two processes together, their sum is y e iht/ h x particle + x e iht/ h y antiparticle = [ iπδ((ct) 2 x y 2 )sign(t) ] (ct) π 2 (6) We see that the principal value part of the expression, which was nonzero outside of the light cone, has canceled. What remains describes the wave function of the initial particle spreading along its light cone, as we might expect for a massless particle, which travels at the speed of light. This has the further interesting consequence. The amplitude for the massless particle to travel at any speed other than that of light vanishes. The upshot of the above development is that a correct treatment of a quantum mechanical particle which also obeys the laws of special relativity requires more than just single particle quantum mechanics. The resolution of the difficulty that we have suggested needs an anti-particle. Quantum field theory will supply us with an anti-particle. Another lesson is that 8

9 the properties of the anti-particle must be finely tuned to be very similar to that of the particle. Otherwise the exact cancellation of the amplitude outside of the light cone would not happen. We will eventually see that this fine-tuning is generally a property of the relativistic wave equations which replace the Schrödingier equation. They have both positive and negative energy solutions which we shall interpret as belonging to the particle and the anti-particle that the wave equation simultaneously describes. We will put off further discussion of this fact until we study wave equations. References [] E.C.C. Stueckelberg, Helvetica Physica Acta, 4, 5 (94); E.C.C. Stueckelberg, Helvetica Physica Acta, 5, 23 (942). [2] R.P. Feynman, Phys. Rev. 74, 939 (948). 9