Introduction to Quantum Field Theory. Matthew Schwartz Harvard University


 Terence Townsend
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1 Introduction to Quantum Field Theory Matthew Schwartz Harvard University Fall 008
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3 Table of contents The Microscopic Theory of Radiation Blackbody Radiation Einstein Coefficients Quantum Field Theory Spontaneous and stimulated emission Lorentz Invariance and Conventions Introduction Conventions dimensional analysis π s Lorentz Invariance rotations Lorentz transformations Discrete transformations Second Quantization Introduction Simple Harmonic Oscillator Plane waves as oscillators Second Quantization more details Normalization and Lorentz Invariance Implications of Second Quantization Classical Field Theory Introduction Classical Field Theory Equations of Motion Currents Coulombs law Classical Fourier transform interlude Coulomb potential again Green s functions Old Fashioned Perturbation Theory Perturbation Theory in Quantum Field Theory Some early infinities Stark Effect Lamb shift a la Oppenheimer OldFashioned Perturbation Theory Examples stark effect
4 4 Table of contents 5.4. Lamb shift a la Oppenheimer Coulomb s law from OFPT Cross Sections and Decay Rates Cross sections Decay Rates Special cases Nonrelativistic limit e + e µ + µ The LSZ reduction formula and the Feynman propagator LSZ reduction theorem Feynman propagator Feynman Rules TimeDependent Perturbation Theory perturbative solution for Dyson s series U relations Ground State meaning of I subscript Summary Time ordered products and contractions Wick s theorem Feynman diagrams symmetry factors signs of momenta Vacuum diagrams Treelevel φ scattering Mandelstam variables With derivative couplings Summary Gauge Invariance Introduction Unitary representations of the Poincare group a simple example Embedding particles into fields spin massive spin massless spin summary Covariant derivatives Conserved Currents Noether s theorem example summary Quantizing spin massive spin massless spin Photon propagator gauges
5 Table of contents gauge invariance Why gauge invariance? Scalar QED Introduction Quantizing Complex Scalar fields Dirac s sea Feynman rules for scalar QED External states Scattering in scalar QED Gauge invariance in scalar QED Lorentz invariance and charge conservation Lorentz invariance for spin Spinors and the Dirac Equation Introduction Representations of the Lorentz Group General representations unitary representations summary Spin representation Lorentz invariants Dirac matrices Rotations Lorentz Invariants Coupling to the photon Weyl spinors Helicity Spin and statistics Majorana fermions summary Solving the Dirac equation free field solutions normalization and spin sums charge conjugation Quantum Electrodynamics Introduction Quantizing Spinors Dirac Propagator Spin statistics Causality QED Feynman rules Example Treelevel QED Matrix identities e + e µ + µ unpolarized scattering differential cross section Rutherford scattering e p + e p QED amplitude
6 6 Table of contents 3.3. corrections to Rutherford Compton scattering KleinNishina formula photon polarization sums tchannel term KleinNishina formula lowenergy limit highenergy behavior singularity cancellation from gauge invariance Processes in QED Path Integrals Introduction Gaussian integrals The Path Integral path integral in QM Path integral in QFT Classical limit Timeordered products Current shorthand Solving the free Path Integral point function Interactions The ward identity Gauge invariance: FadyeevPopov Fermionic path integral SchwingerDyson equations Ward identity from SchwingerDyson equations BRST invariance The Casimir Effect Introduction Casimir effect Hard Cutoff Regulator Dependence heat kernel regularization other regulators regulator independent derivation string theory aside Counterterms Scalar field theory example renormalizion of λ counterterm interpretation Regularization Schemes and Tricks Feynman Parameters Wick Rotations Quadratic integrals derivative method PauliVillars regularization Dimensional Regularization k µ integrals Summary
7 Table of contents 7 7 Vacuum Polarization Introduction Scalar φ 3 theory interpretation Vacuum polarization in scalar QED Spinor QED Physics of vacuum polarization small momentum Lamb shift large momentum logarithms of p running coupling Other graphs g Introduction Vertex correction Evaluating the graphs Systematic Renormalization Measuring Correlation Functions Tadpoles Electron selfenergy selfenergy loop graph wavefunction renormalization at m e = wavefunction renormalization with a mass oneparticle irreducible graphs polemass renormalization conditions Summary of electron selfenergy renormalization Amputation Renormalized perturbation theory point function in renormalized perturbation theory Photon selfenergy point function: vacuum polarization gauge invariance and counterterms point functions QED renormalization conditions Minimal Subtraction propagator in MS δ = δ : implications and proof charge is not renormalized Allorders proof of δ = δ Infrared Divergences Introduction Total cross section for e + e µ + µ Loops Tree level graph Vertex Correction counterterms expanding out the process full e + e µ + µ Realemission: Final state radiation Initial state radiation
8 8 Table of contents 0.6 Applications of infrared divergences e + e µ + µ in dim reg loops dimensionless integrals in dimreg total virtual part real emission Compton scattering The Lamb Shift Hydrogen atom spectrum summary Bethe s Lamb shift calculation Lamb shift phenomenology Bethe s log spinorbit coupling Renormalizability of QED Introduction Renormalizability of QED point functions ,6,... point functions renormalizability to all orders Nonrenormalizable field theories nonrenormalizable theories are finite too NonRenormalizable and SuperRenormalizable Field Theories Introduction Schrodinger Equation The 4Fermi theory quantum predictions of the Fermi theory UVcompleting the Fermi theory Theory of Mesons Quantum Gravity Predictions of gravity as a nonrenormalizable field theory Summary of nonrenormalizable theories Quantum predictions which are not logs from physics to philosophy Superrenormalizable field theories tadpoles relevant interactions scalar masses fermion and gauge boson masses summary Implications of Unitarity Introduction The Optical Theorem Decay rates Cutting Rules Propagators and polarization sums BreitWigner Cross sections: The Froissart Bound
9 Chapter The Microscopic Theory of Radiation. Blackbody Radiation Quantum Mechanics began on October 9, 000 with Max Planck s explanation of the blackbody radiation spectrum. Although his result led to the development of quantum mechanics, that is, the quantum mechanics of electrons, Planck s original observation was about the quantum nature of light, which is a topic for quantum field theory. So, radiation is a great way to motivate the need for a quantum theory of fields. In 900 people were very confused about how to explain the spectrum of radiation from hot objects. The equipartition theorem says that a body in thermal equilibrium should have energy equally distributed among the various modes. For a hot gas, this gives the MaxwellBoltzmann distribution of thermal velocities, which is beautifully confirmed by data. But when you try to do the same thing for light, you get a very strange result. Take a blackbody, which we can think of as a hot box of light (also known as a Jeans cube) of size L. We know that there are standing electromagnetic waves inside with frequencies ν n = π nk L for integer 3vectors nk. Before 900 people thought you can have as much or as little energy in each mode as you want. Then the equipartition theorem says that since there are an infinite number of modes excited, the blackbody should emit light equally in all frequencies. In particular, there should be a lot of UV light the ultraviolet catastrophe. Instead, the distribution looks like a MaxwellBoltzmann distribution. This led Planck to postulate that the energy of each electromagnetic mode in the cavity is quantized in units of frequency (.) E n =hν n = π L h nk = kkn (.) where h is Planck s constant. Einstein then interpreted this as saying light is made up of particles, called photons. Note that if the excitations are particles, then they are massless m n =E n kkn =0 (.3) Thus Planck and Einstein said that light is massless photons. Light having energy leads to the photoelectric effect; light having momentum leads to Compton scattering. With Planck s energy hypothesis, the thermal distribution is easy to compute. Each mode of frequency ν n can be excited j times, giving energy E n = jν n in that mode. The probability of finding that much energy in the mode is the same as the probability of finding a certain amount of energy in anything, P(E) = exp( E/k B T). Thus the expectation value of the the energy in each mode is E n = j=0 j=0 (je n )e jenβ e jenβ = d dβ e hνnβ = hν n e hνnβ e hνnβ (.4) 9
10 0 The Microscopic Theory of Radiation where β = /k B T. (This derivation is due to Debye. Nowadays we do the calculation using a canonical ensemble.). Now let s take the continuum limit, L. The distribution of energy in a blackbody becomes Thus, E = d 3 hν n n e hνnβ = 4πh d nk = 4πh L3 8π 3 dν I(ν) = V ν 3 e hνβ de dν = h ν 3 π e hνβ nk ν n e hνnβ Which is Plank s showed in 900 correctly matches experiment. (NB: I threw in a factor of for the polarizations of light, which you shouldn t worry about at all yet.). What does this have to do with Quantum Field Theory? Well, in order for any of this to make sense, light has to be able to equilibrate. If we shine monochromatic light into the box, eventually it reaches all frequencies. But if different frequencies are different particles, these particles have to get created by something. Quantum field theory tells us how radiation works at the microscopic level. (.5) (.6) (.7). Einstein Coefficients The easiest handle on the creation of light comes from the coefficient of spontaneous emission. This is the rate for an excited atom to emit light. This phenomenon had been observed in chemical reactions, and as a form of radioactivity. But it was only understood statistically. In fact, Einstein in 906 developed a beautiful proof of the relation between emission and absorption based on the existence of thermal equilibrium. Einstein said: Suppose we have a cavity full of atoms with energy levels E and E. Let ω = E E. In equilibrium the number densities are determined by Boltzmann distributions. n =Ne βe n = Ne βe (.8) The probability for an E atom to emit a photon of energy ω and transition to state E is called the coefficient for spontaneous emission A. The probability for a photon of energy ω to induce a transition from to is proportional to the coefficient of stimulated emission B. The induced rate is also proportional to the number of photons of energy ω. Then dn =[A + B I(ω)] n (.9) The probability for a photon to induce a transition from to is called the coefficient of stimulated absorption B. Then dn = B I(ω)n (.0) Now if we assume the system is in equilibrium then the rate going up must be the same as the rate going down: dn = dn. Then [A+B I(ω)] n =B I(ω)n (.) [ ] B e βe B e βe I(ω) =Ae βe (.) But we already know that I(ω) = A B e βω B (.3) I(ω)= h π ω 3 e βω (.4)
11 .4 Spontaneous and stimulated emission from above. Since equilibrium must be satisfied at any temperature, and B = B = B (.5) A B = π ω3 (.6) This is a beautiful proof. If light has energy ω with a Planck distribution, then equilibrium relates A and B. However, why should you need equilibrium to say this? What causes an atom to emit a photon the first place? The B coefficients, for stimulated emission/absorption can be calculated with a background electromagnetic field and quantum mechanics. Dirac did this in 93. However, how do you calculated the coefficient of spontaneous emission B? It took 0 years to be able to calculate A/B from first principles. It took the invention of Quantum Field Theory..3 Quantum Field Theory The basic idea behind the calculation of the spontaneous emission coefficient was to treat the photon excitations like a quantum mechanics problem. Start by just looking at a single photon mode at a time, say of energy. This mode can be excited j times. Each excitation adds energy to the system. There is a quantum mechanical system with this property one you may remember from your quantum mechanics course: the simple harmonic oscillator. We will review it for homework. For now, let me just remind you of how it works. The easiest way to study the system is with creation and annihilation operators. These satisfy: There is also the number operator which counts modes Then, and so Where the normalization is set so that [a, a ] = (.7) N =a a (.8) N n =n n (.9) Na n = a aa n =a n +a a a n = (n + a)a n (.0) a n = n + n + (.) a n = n n (.) n n = (.3) Now for the photon, we want to be able to arbitrarily excite any mode in the cavity. Thus N = {k} a k a k (.4) This seems like absolutely nothing just algebra. But it s not..4 Spontaneous and stimulated emission For example, let s look how a photon interacts with an atom. We want to use Fermi s golden rule to calculate the transition rate between states. Recall that the rule states that the rate is proportional to the matrix element squared Γ M δ(e f E i ) (.5)
12 The Microscopic Theory of Radiation where the δfunction serves to enforce energy conservations. Using perturbation theory, we know we can write interactions of the form M = f H int + i (.6) We don t know exactly what the interaction Hamiltonian H int is, but it s got to have something to connect the initial and final atomic states and some creation operator or annihilation operator to create the photon. So it must look like H int = H 0 a k + H 0 a k (.7) For the transition, the initial state is an excited atom with n photons of momenta k: The final state is a lower energy atom with n + photons of momentum k i = atom ; n k (.8) So, where Thus f = atom; n k + (.9) M = atom; n k + H 0 a k +H 0 a k atom ; n k = atom H 0 atom n k + a k n k =M 0 n k + n k + n k + =M 0 n k + (.30) (.3) (.3) (.33) M 0 = atom H 0 atom (.34) M = M 0 (n k + ) (.35) But if we are exciting an atom, then the initial state is an unexcited atom and n photons: and the final state we have lost a photon and have an excited atom i = atom; n k (.36) Then Thus f = atom ; n k (.37) M = atom ; n k H 0 a k + H 0 a k atom; n k (.38) = atom H 0 atom n k a k n k (.39) =M 0 n k (.40) dn = M 0 n (n k + ) (.4) Remember Einstein s equations: dn = M 0 n (n k ) (.4) dn = (A+B I(ω))n (.43) dn = B I(ω)n (.44)
13 .4 Spontaneous and stimulated emission 3 Now, E = d 3 k (π) 3(hk)n k = dk hk3 π n k (.45) I(k)= de dk = hk3 π n k (.46) n k = π h I(k)k3 (.47) Note that this doesn t depend on what n k is, this is just a phase space integral. There is no mention of temperature or of equilibrium it is true for any values of n k. Thus [ ] π dn =Cn k 3I(k)+ (.48) Thus π dn = Cn k3i(k) (.49) B = B (.50) A B = π k 3 (.5) Which are Einstein s relations. Beautiful! This derivation is due to a paper of Dirac s from 97. Note that we never needed to talk about statistical mechanics, ensembles, or temperature.
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15 Chapter Lorentz Invariance and Conventions. Introduction This lecture is meant to establish the conventions used in the course, and to review Lorentz invariance and notation. It is taken partly from Mike Luke s QFT lectures http: // luke/phy403/references_files/lecturenotes.pdf (.) In particular section B. You might want to look at those lectures for more, and somewhat different, details.. Conventions.. dimensional analysis We will set = h = (.) π c = (.3) throughout. This makes all quantities have dimensions of mass to some power. Sometimes we write the mass dimension of a quantity with brackets: [M d ] = d (.4) Really, everything is either a length (eg position x) or a mass (eg momentum p). With = c = [p] = / [x] =. So [x] = (.5) [t] = (.6) Thus Action is dimensionless So Lagrangians have dimension 4 [p] = (.7) [veolcity] = [x] [t] = 0 (.8) [d 4 x] = 4 (.9) [S] = [ d 4 x L] = 0 (.0) [L] = 4 (.) 5
16 6 Lorentz Invariance and Conventions For example, a free scalar field has Lagrangian thus and so on... π s L = φd φ (.) [φ] = (.3) Keeping the factors of π straight can be a pain. They come from going back and forth between position and momentum space. For example, dke ikx =πδ(x) (.4) In fact, you should internalize this relation as you will use it a lot. The convention we will use is that momentum space integrals have πs and a + sign in the exponent d f(x) = 4 k (π) 4 f (k)e ikx (.5) The position space integrals have no π s and a minus sign in the exponent f (k)= d 4 xf(x)e ikx (.6).3 Lorentz Invariance Lorentz invariance implies a symmetry under rotations and boosts..3. rotations Rotations you should be extremely familiar with, and are more intuitive. For example, under rotations, a vector (x, y) transforms as x x cosθ + y sinθ (.7) y x sinθ + y cosθ (.8) We can write this as ( x y Or as ) ( )( cosθ sinθ x sinθ cosθ y x i R ij x j, x i = ) ( x cosθ + y sinθ = xsinθ + y cosθ ( x y ) (.9) ), i=, (.0) We separate upper and lower indices to refer to column and row vectors respectively. So we can also write ( ) cosθ sinθ x i = (x, y) (x, y) =x sinθ cosθ i T R ij (.) Note that R T = R. That is or equivalently R T ij R jk = δ jk =( ) (.) R T R = (.3)
17 .3 Lorentz Invariance 7 This is actually enough to characterize it as a rotation, and we don t need the explicit form of the rotation matrix as a function of θ anymore. We can contract upper and lower indices ( ) x x i x i =(x, y) = x y + y (.4) This is just the norm of the vector x i and is rotation invariant. That x i x i means sum over i is sometimes called the Einstein summation convention. In terms of matrices x i x i x i R T Rx k =x i x i (.5) since R T = R. In fact, we can formally describe the rotation group as preserving the inner product x i x i =δ ij x i x j R ikr jlδ kl =[(R T )(δ)(r)] ij = (R T R) ij = δ ij (.6) which we can check explicitly.3. Lorentz transformations Lorentz transformations work exactly like rotations, except we use vectors with 4 components with a Greek index V µ, and the matrices satisfy Λ T ηλ = η = (.7) η µν is known as the Minkoswki metric. So Lorentz transformations preserve the Minkowskian inner product x i x j = x i η ij x j = x 0 x x x 3 (.8) The Lorentz group is the most general set of transformations with this property. A general Lorentz transformation can be written as a combination of 3 rotations and 3 boosts: Λ = cos θ xy sin θ xy 0 sin θ xy cosθ xy coshβ x sinhβ x sinhβ x coshβ x cosθ xz sin θ xz cosθ xz sin θ xz sin θ xz cosθ xz sin θ xz cos θ xz coshβ y sinhβ y coshβ z sinhβ z sinhβ y coshβ y sinhβ z coshβ z (.9) (.30) Note that these matrices do not commute, so the order we do the rotations and boosts is important. We will only very rarely need an actual matrix representation of the group elements like this, but it is helpful to see it. So now vectors transform as V µ Λ µ ν V ν (.3) W µ W ν (Λ T ) ν µ (.3) That is, these vectors transform covariantly or contravariantly. The contraction of a covariant and contravariant vector is Lorentz invariant. W µ V µ W ν (Λ T ) ν µ Λ µ α V α =W µ V µ (.33)
18 8 Lorentz Invariance and Conventions Only when you are using explicit representations of the Lorentz group elements to the index positions really matter. Otherwise as long as the indices are contracted, I don t care whether they are up or down: V µ W µ = V µ W µ = V µ W µ (.34) =V 0 W 0 V W V W V 3 W 3 (.35) This is a bad habit and I probably shouldn t be encouraging it, but as I said, this is meant to be a practical course. Conventionally, position vectors have upper indices derivatives have lower indices momenta are like derivatives and have lower indices x µ = (t, x, y, z) (.36) µ = x µ = ( t, x, y, z ) (.37) p µ = (E, p x, p y, p z ) (.38) Note that sometimes you will see (Mike Luke does this) p µ = (E, p x, p y, p z ). It doesn t matter if you put these signs in the individual vector elements as long as you always contract with the Minkowski metric η = diag(,,, ). That is, the signs only really matter when you write out the explicit contraction of a Lorentz invariant expression. Scalar fields are Lorentz invariant φ(x) φ(x) (.39) Note that our definitions of x may be changing x Λx, but the spacetime point x is fixed, just our labels for it change. For example, think of temperature as a scalar. If I change frames, the labels for the points changes, but the temperature at that point stays the same. Vectors transform as V µ (x) Λ µν V ν (x) (.40) I m not bothering with up and down indices, but the proper thing to do is write V µ = Λ µ ν V ν. Again, x transforms too, but passively. The difference with a scalar is that the components of a vector field at the point x rotate into each other as well. Tensors transform as and so on. We also write Note that you should never have anything like T µν Λ µα Λ νβ T αβ (.4) = µ = µ µ = t x y z (.4) = = x + y + z (.43) V µ W µ X µ (.44) with 3 (or more) of the same indices. Be very careful to relabel things. For example, don t write instead write (V )(W ) =V µ V µ W µ W µ (.45) (V )(W )=V µ W ν =V µ V µ W ν W ν = η µα η νβ V µ V α W ν W β (.46) You will quickly get the hang of all this contracting.
19 .4 Discrete transformations 9 In summary, we say V = V µ V µ, φ,, µ V µ (.47) are Lorentz invariant. But V µ, F µν, µ, x µ (.48) are Lorentz covariant (or contravariant, but only mathematicians say that)..4 Discrete transformations Lorentz transformations are those which preserve the Minkowski metric Equivalently, they are those which leave invariant. So the transformations and Λ T ηλ = η (.49) x µ x µ =t x y z (.50) P: (t, x, y, z) (t, x, y, z) (parity) (.5) T: (t, x, y, z) ( t, x, y, z) (time reversal) (.5) are technically Lorentz transformations too. They can be written as T = (.53) P = (.54) We say that the Lorentz transformations which are continuously connected to the identity transformation are Proper Othonchronous Lorentz transformations. They can be characterized by detλ= (.55) Λ 00 0 (.56) With T and P we get all Lorentz transformations. Note detp = dett = and T 00 < 0. Also, we say a vector is timelike when and spacelike when V µ > 0, (timelike) (.57) V µ <0, (spacelike) (.58) Obviously time = (t, 0, 0, 0) is timelike and (0, x, 0, 0) is spacelike. Whether something is timelike or spacelike is preserved under Lorentz transformations since the norm is preserved. If it s norm is zero we say it is lightlike V µ = 0, (lightlike) (.59) If V µ is a 4momentum, this says that it is massless. Photons are massless, which is the origin of the term lightlike.
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21 Chapter 3 Second Quantization 3. Introduction Last time we saw that if we treat each mode of the photon as a separate particle, and give it multiple excitations like it s a harmonic oscillator, we can derive the relation between the coefficients of induced and spontaneous emission without resorting to statistical mechanics. This was our first QED calculation. Today we will make this a little more precise, give it a little more justification, and pave the way for more detailed computations in Quantum Electrodynamics. I should mention here that there are two ways of quantizing a field theory. The first is the called canonical quantization. It is historically how quantum field theory was understood, and closely follows what you learned in quantum mechanics. It s also easy to see the relation between the formalism and physical problems. The second way is called the Feynman Path Integral. It is more beautiful in a way, and more general, but it is hard to understand physically what you are calculating. In any case, both are necessary for a proper education in quantum field theory, so we will start with the traditional canonical quantization. From now on I will set = c =. I will also be cavalier about signs and small numerical factors. This is mostly because I am too lazy to make sure they are right. But it is also because I want to keep talking about electromagnetic fields, which are physical, and to keep treating them like scalars, which are easy to study. It turns out that if you worry about the signs and treat the photon like a scalar, things just don t work (after all, that s how we know the photon is not a scalar!). Eventually, we will to QED properly and then I will be careful about signs, but not yet. 3. Simple Harmonic Oscillator Recall the simple harmonic oscillator. Anything with a linear restoring potential (the simplest thing possible), like a spring, or a string with tension, or a wave, is a harmonic oscillator. For example, a spring has m d x dt kx=0 x(t)=cos k t (3.) m I.e. it oscillates with frequency The classical Hamiltonian for this system is ω = k m (3.) x(t) =c e iωt + c e iωt (3.3) To quantize it, we say H = p m mω x (3.4) [x, p] = ih (3.5)
22 Second Quantization A very very important trick is to write this as a = mω ( x + ip ) mω with Then, a = mω ( x ip ) mω (3.6) [a, a ] = (3.7) ( H = ω a a + ) Thus energy eigenstates are eigenstates of the number operator which is Hermetian. Also, all the stuff we saw yesterday is true. (3.8) N = a a (3.9) N n =n n (3.0) a n = n + n + (3.) a n = n n (3.) One more useful fact is how these operators evolve in time. Recall that in the Heisenberg picture, states don t evolve, operators do. Then i d ( [a, dt a =[a, H] = ω a a + )] = ω(aa a a aa)= ω[a, a]a = ωa (3.3) a(t)=e iωt a(0) (3.4) 3.3 Plane waves as oscillators The excitations of a plane electromagnetic wave in a cavity are very similar to the excitations of a harmonic oscillator. Recall that there s a nice Lorentzcovariant treatment of electromagnetism using 0 E x E y E z E F µν = x 0 B z B y (3.5) E y B z 0 B y E z B y B x 0 F is antisymmetric. This makes Maxwell s equations have the simple form (in empty space) µ F µν = 0 (3.6) The nice thing about using F µν is that it concisely encodes how E and B rotate into each other under boosts. We can always write with F µν = µ A ν ν A µ (3.7) µ A µ =0 (3.8) Requiring µ A µ = 0 is not necessary. It is a gauge choice (Lorentz gauge). We will discuss gauge invariance in great detail in a few weeks. For now, it is enough to know that the physical E and B fields can be combined in this way.
23 3.4 Second Quantization 3 Then, µ F µν = A ν ν ( µ A µ )= A ν = 0 (3.9) where A(x)=a k (t)e i kkxk ( t + ω k )a k (t)=0, ω k = kk (3.) From here on, I will drop the subscript on A ν, writing it as A. You can keep the ν if you want, but none of the qualitative conclusions that follow today will depend on it, and it makes things less cluttered. So, ( t x )A=0 (3.0) The solutions are plane waves. For example, one solution is (3.) This is exactly the equation of motion of a harmonic oscillator. Thus electromagnetic fields are oscillators. We can decompose a general solution to Maxwell s equations as d A(x, t)= 3 k [ ak (π) 3 (t)e ikx +a k (t)e ] ikx (3.3) 3.4 Second Quantization Since the modes of an electromagnetic field have the same classical equations as a simple harmonic oscillator, we can quantize them the same way d H = 3 k (π) 3 ω k(a k a k + ) (3.4) This is known as second quantization. First quantizing is saying that all the modes have energy proportional to their frequency. Second quantizing is saying that there are integer numbers of excitations of each of these modes. But this is somewhat misleading the fact that there are discrete modes is a classical phenomenon. The two steps really are ) interpret these modes as having momenta k = hν and ) quantize each mode as a harmonic oscillator. In that sense we are only quantizing once. But these are only words. There are two new features in second quantization. We have many quantum mechanical systems one for each k all at the same time.. We interpret the n th excitation of the k harmonic oscillator as having n photons. Let s take a moment to appreciate this second point. Recall the old simple harmonic oscillator the electron in a quadratic potential. We would never interpret the n states of this system as having n electrons, because that s completely wrong. (Actually, there is an interpretation of these n states as n particles called phonons. Phonons are a special topic which may be covered in semester.) The fact that a pointlike electron in a quadratic potential has analogous equations of motion to a Fourier component of the electromagnetic field is just a coincidence. Don t let it confuse you. Both are just the simplest possible dynamical systems, with linear restoring forces. To set up a proper analogy we need to first treat the electron as a classical field (we don t know how to do that yet), and find a set of solutions (like the discrete frequencies of the EM waves). Then we would quantize each of those solutions, allowing n excitations. However, if we did this, electrons would have BoseEinstein statistics. Instead, they must have FermiDirac statistics, so restrict n = 0,. There s a lot to learn before we can do electrons, so let s stick with photons for now.
24 4 Second Quantization 3.4. more details Now let s get a little more precise about what this Hamiltonian means. The natural generalization of is These a k operators create particles with momentum k [a, a ] = (3.5) [a k, a p ] = (π) 3 δ 3 (p k) (3.6) a k 0 = k (3.7) ω k This factor of ω k homework that it has nice Lorentz transformation properties. Also, we know how the momentum states k evolve from the way a k evolves: For normalization, we start with Then And a complete set is So that k = is just a convention, but it will make some calculations easier. You will show for k, t =a k (t) 0 =e iωkt a k (0) 0 =e iωkt k (3.8) 0 0 = (3.9) p k = ω p ω k 0 a p a k 0 =ω p (π) 3 δ 3 (p k) (3.30) = d 3 p (π) 3 p p k = p ω p d 3 p (π) 3 p p (3.3) ω p d 3 p (π) 3 ω p (π) 3 δ 3 (p k)= k (3.3) ω p Finally, define A(x)= d 3 k ] (π) 3 [a k e ikx +a ω k e ikx k (3.33) This is the same as the free solutions to Maxwell s equations, equation (3.3) but instead of a k being a function, it is now the creation/annihilation operator for that mode. The factor of ω k has been added for later convenience, and in fact guarantees the A(x) is Lorentz invariant. The connection with Eq. (3.3) should be considered only suggestive, and this taken as the definition of an operator A(x) constructed from the creation and annihilation operators a k and a k. Note that: d p A(x) 0 = 0 ω 3 k ] pap (π) 3 [a k e ikx + a ω k e ikx 0 (3.34) k d = 3 k ω [ ] p (π) 3 e ikx 0 a p a k 0 +e ikx 0 a p a ω k 0 (3.35) k So That is A(x) creates a photon at position x. =e ikx (3.36) A(x) 0 = x (3.37) p x =e ikx (3.38)
25 3.6 Implications of Second Quantization Normalization and Lorentz Invariance The important formulas are [a k, a p ] = (π) 3 δ 3 (p k) (3.39) Where A(x)= a k 0 = k (3.40) ω k d 3 k ] (π) 3 [a k e ikx +a k e ikx ω k ω k kk +m (3.4) (3.4) All these ω k factors are a conventional normalization, designed to make things easily Lorentz invariant. They come from dk 0 δ(k m )θ(k 0 )= ω k (3.43) which implies d 3 k ω k (3.44) is Lorentz invariant. (You will check these relations in Problem Set ). It also implies is Lorentz invariant. Then, ω k δ 3 (p k) (3.45) p k = ω k ω p 0 a p a k 0 =ω k (π) 3 δ 3 (p k) (3.46) is Lorentz invariant, which is the whole point. Also, d A(x) 0 = 3 k (π) 3 e i pk xk p (3.47) ω p So we this same Lorentz invariant combination d3 k ω p appears again. 3.6 Implications of Second Quantization All we have done to quantize the electromagnetic field is to treat it as an infinite set of simple harmonic oscillators, one for each wavenumber k. So Quantum Field Theory is just quantum mechanics with an infinite number of fields. Even if we put the system in a box of size L, so that k = π n, we can still have n. This will L become a tremendous headache for us, since these s will creep into practically every calculation we do. A big chunk of this course will be devote to understanding them, and at some level they are still not understood. In quantum mechanics we were used to treating a single electron in a background potential V (x). Now that background, the electromagnetic system, is dynamical, so all kinds of funny things can happen. We already saw one, Dirac in his calculation of the Einstein coefficients. We can be a little bit more explicit about how we got this now. We write the Hamiltonian as H = H 0 + H int (3.48) Where H 0 is some state that we can solve exactly. In this case it is the atom + the free photons H 0 =H atom +H photon (3.49)
26 6 Second Quantization The H int is some modification that is small enough to let us use perturbation theory. Then all we need is Fermi s golden rule, the same one from quantum mechanics (which you might want to review). It says the rate for transitions between two states is proportional to the square of the matrix element of the interaction between the two states. Γ = f H int i δ (E f E i ) (3.50) In our case, H atom gives the states ψ n of the hydrogen atom, with energies E n. H photon is the Hamiltonian above, d H photon = 3 k (π) 3ω k(a k a k + ) (3.5) and we can treat the interaction semiclassically: H int = eaj atom (3.5) Here J atom is shorthand for the atom treated as a source for photons. Essentially, we are treating the atom as a fixed background that is, we won t need to consider how the atom gets deformed. The important point is that this interaction has an A field in it. Then, according to Fermi s golden rule, the transition probability is proportional to the matrix element of the interaction squared M = atom ; n k H int atom; n k = atom J atom atom n k (3.53) M = atom; n k + H int atom ; n k = atom J atom atom n k + (3.54) where we have used So, n k A n k = n k + A n k = d 3 p (π) 3 n k a p n k = n k d 3 p (π) 3 n k + a p n k = n k + (3.55) (3.56) dn M = atom J atom atom n k (3.57) dn = M = atom J atom atom (n k + ) (3.58) This was derived by Dirac in his calculation of the Einstein coefficients. Note that we only used one photon mode, of momentum k, so this was really just quantum mechanics. All QFT did was give us a deltafunction from the d 3 p integration.
27 Chapter 4 Classical Field Theory 4. Introduction We have now seen how Quantum Field Theory is defined as quantum mechanics with an infinite number of oscillators. QFT can do some remarkable things, such as explain spontaneous emission without statistical mechanics. But it also seems to lead to absurdities, such as the infinite shift in the energy levels of the Hydrogen atom. To show that QFT is not absurd, but extremely predictive, we will have be very careful about how we do calculations. We ll begin by going through carefully some of the predictions which QFT gets right without infinities. These are the treelevel processes, which means they are leading order in perturbation theory. We will start to study them today. 4. Classical Field Theory Before we get in to more QFT calculations, let s review some elements of classical field theory. The Hamiltonian is the sum of the kinetic and potential energies of a system H = K +V (4.) So the Hamiltonian gives the total energy. The Hamiltonian has all the information about the time evolution of a system. This is true both classically, where timetranslation is determined by a Poisson bracket, and quantum mechanically, where timetranslation is determined by a commutator. The problem with Hamiltonians is that they are not Lorentz invariant. The Hamiltonian picks out energy, which is not a Lorentz scalar, but the 0 component of a Lorentz vector: V µ = (H, PK). It is great for nonrelativistic systems, but for relativistic systems we will almost exclusively use Largrangians. The Lagrangian (for fields, the Lagrangian density) is the difference between kinetic energy and potential energy L =K V (4.) So while the Hamiltonian corresponds to a conserved quantity, energy, the Lagrangian does not. Dynamics from a Lagrangian system is determined by trying to minimize the action, which is the integral over space and time of the Lagrangian density S = d 4 x L(x) (4.3) We don t usually talk about kinetic and potential energy in quantum field theory. Instead we talk about kinetic terms and interactions. Kinetic terms are bilinear in the fields. Bilinear means there are two fields. So kinetic terms look like L K φ φ, ψ ψ, 4 F µν, m φ, (4.4) where φ φ, φ µ A µ, F µν = µ A ν ν A µ (4.5) 7
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