2012 Matthew Schwartz I-4: Old-fashioned perturbation theory
1 Perturbative quantum field theory
The slickest way to perform a perturbation expansion in quantum field theory is with Feynman diagrams. These diagrams will be the main tool we’ll use in this course, and we’ll derive the dia- grammatic expansion very soon. Feynman diagrams, while having advantages like producing manifestly Lorentz invariant results, give a very unintuitive picture of what is going on, with particles that can’t exist appearing from nowhere. Technically, Feynman diagrams introduce the idea that a particle can be off-shell, meaning not satisfying its classical equations of motion, for 2 2 example, with p m . They trade on-shellness for exact 4-momentum conservation. This con- ceptual shift was critical in allowing the efficient calculation of amplitudes in quantum field theory. In this lecture, we explain where off-shellness comes from, why you don’t need it, but why you want it anyway. To motivate the introduction of the concept of off-shellness, we begin by using our second quantized formalism to compute amplitudes in perturbation theory, just like in quantum mechanics. Since we have seen that quantum field theory is just quantum mechanics with an infinite number of harmonic oscillators, the tools of quantum mechanics such as perturbation theory (time-dependent or time-independent) should not have changed. We’ll just have to be careful about using integrals instead of sums and the Dirac δ-function instead of the Kronecker δ. So, we will begin by reviewing these tools and applying them to our second quantized photon. This is called old-fashioned perturbation theory (OFPT). As a historical note, OFPT was still a popular way of doing calculations through at least the 1960s. Some physicists, such as Schwinger, never embraced Feynman diagrams and continued to use OFPT. It was not until the 1950s though the work of Dyson and others that it was shown that OFPT and Feynman diagrams gave the same results. Despite the prevalence of Feynman’s approach in modern calculations, and the efficient encapsulation by the path integral formalism (Lectures II-7 and onward), OFPT is still worth understanding. It provides complementary physical insight into quantum field theory. For example, a souped-up version of OFPT is given by Schwinger’s proper time formalism (Lecture IV-9), which is still the best way to do certain effective action calculations. This lecture can be skipped without losing continuity with the rest of the text.
2 Lippmann-Schwinger equation
Just like in quantum mechanics, perturbation theory in quantum field theory works by splitting the Hamiltonian up into two parts
H = H0 + V (1) where the eigenstates of H0 are known exactly, and the potential V gives corrections which are small in some sense. The difference from quantum mechanics is that in quantum field theory the states often have a continuous range of energies. For example, in a Hydrogen atom coupled to an electromagnetic field the associated photon energies E = ωk = kQ can take any values. Because of the infinite number of states, the methods look a little different, but we will just be doing the natural continuum generalization of perturbation theory in quantum mechanics.
1 2 Section 2
We are often interested in a situation where we know the state of a system at early times and would like to know the state at late times. Say the state has a fixed energy E at early and late times (of course, it’s the same E). There will be some eigenstate of H0 with energy E, call it | φ i . So,
H0| φ i = E | φ i (2) If the energies E are continuous then we should be able to find an eigenstate | ψ i of the full Hamiltonian with the same eigenvalue H | ψ i = E| ψ i (3) and we can formally write 1 | ψ i = | φ i + V | ψ i (4) E − H0 which is trivial to verify. This is called the Lippmann-Schwinger equation. 1 The inverted object appearing in the Lippmann-Schwinger equation is a kind of Green’s function known as the Lippmann-Schwinger kernel 1 ΠLS = (6) E − H0
The Lippmann-Schwinger equation is useful in scattering theory where | φ i are free states, such as momentum eigenstates with continuous energies, at early and late times, and the poten- tial acts at intermediate times to induce transitions among these states. It says the full wave- function | ψ i is given by the free wavefunction | φ i plus a scattering term. What we would really like to do is express | ψ i entirely in terms of | φ i . Thus we define an operator T by V | ψ i = T| φ i (7) T is known as the transfer matrix. Inserting this definition turns the Lippmann-Schwinger equation into 1 | ψ i = | φ i + T| φ i (8) E − H0 which formally gives | ψ i in terms of | φ i . Multiplying this by V and demanding the two sides be equal when contracted with any state h φ j | gives an operator equation for T 1 T = V + V T (9) E − H0 We can then solve perturbatively in V to get 1 1 1
T =V + V V + V V V + E − H0 E − H0 E − H0 (10)
=V + V ΠLS V +V ΠLS V ΠLS V +
If we insert complete set j | φ j ih φ j | of eigenstates | φ j i of H0 the matrix elements P 1
h φi | T| φ f i = h φi | V | φ f i + h φ f | V | φ j ih φ j | V | φi i + (11) E − H0
Writing Tfi = h φ f | T| φi i and Vij = h φi | V | φ j i this becomes
Tfi = Vfi + VfjΠLS ( j) Vji + VfjΠLS ( j) Vjk ΠLS ( k) Vki + (12)
1 where ΠLS ( k) = , where again, E = Ei = Ej is the energy of the initial and final state we E − Ek are interested in. This expansion is old-fashioned perturbation theory.
1. Formally, the inverse of E − H0 is not well-defined. Since E is an eigenvalue of H0, det ( E − H0 ) = 0 and − 1 ( E − H0 ) is singular. To regulate this singularity, we can add an infinitesimal imaginary factor iε, leading to 1 | ψ i = | φi + V | ψ i (5) E − H0 + iε with the understanding that ε should be taken to zero at the end of the calculation. Lippmann-Schwinger equation 3
Eq. (12) says that to calculate transitions in perturbation theory there is a sum of terms. In each term the potential creates an intermediate state | φ j i which propagates with the propagator ΠLS ( j) until it hits another potential, where it creates a new field | φk i which then propagates and so on, until they hit the final potential factor, which transitions it to the final state. There is a nice diagrammatic way of drawing this series, called Feynman graphs, which we will see through an example in a moment and in more detail in upcoming lectures. For some termi- nology, the first term, Vij gives the Born approximation (or first Born approximation), the second term the second Born approximation and so on. See for example [Sakuraii chapter 7] for applications of the Lippmann-Schwinger approximation in non-relativistic quantum mechanics.
2.1 Coulomb’s law revisited The example we will compute is one we will revisit many times: an electron scattering off another electron in QED. The transition matrix element for this process is given by the Lipp- mann-Schwinger equation as 1
Tfi = Vfi + Vfn Vni + (13) Ei − En Xn
Here Ei are the initial energy (which is the same as the final energy), and En is the energy of 1 2 the intermediate state. The initial and final states each have two electrons | i i = | ψe ψe i and 3 4 1 1 h f | = h ψe ψe | where the superscripts label the momenta, i.e. ψe has Qp , etc. The intermediate state can be anything in the whole Fock space, but only certain intermediate states will have non-vanishing matrix elements with V. In relativistic field theory, the instantaneous action-at-a-distance of Coulomb’s law is replaced by a process where two electrons exchange a photon which travels only at the speed of light. Thus there should be photons in the intermediate state. Ignoring the spin of the electron and the photon, the interaction of the electron with the photon field can be written as
1 3 V = e d xψe( x) φ( x) ψe ( x) (14) 2 Z
This interaction is local, since the fields ψe ( x) and φ( x) , corresponding to the electrons and 1 photon are all evaluated at the same point. The factor of 2 comes from ignoring spin and treating all fields as representing real scalar particles.
This interaction can turn a state with an electron of momentum Qp1 into a state with an elec- Q tron with momentum pQ 3 and a photon with momentum pγ. Since initial and final states both have two electrons and no photons, the leading term in Eq. (13) vanishes, Vfi = 0. To get a non-zero matrix element, we need an intermediate state | n i with a photon. There are two intermediate states which can contribute. In the first, the photon is emitted from the first electron and the intermediate state is before that photon hits the second electron. We can draw a picture representing this process:
− − e1 time→ e3
~p3
~p1
(15)
~p2
~p4 ¡ − − e2 e4
The vertical dashed line indicates the time at which the intermediate state is evaluated. The second electron feels the effect of a photon that the source, the first electron, emitted at an ear- lier time. We say that the electron states interact in this case through a retarded propagator. 3 γ 2 γ For this retarded case, | n i = | ψe φ ψe i where | φ i is a photon state with momentum pγ. Then,
( R) 3 γ 2 1 2 3 γ 1 2 2 3 γ 1 Vni = e h ψe φ ψe | V | ψe ψe i = e h ψe φ | V | ψe ih ψe | ψe i = e h ψe φ | V | ψe i (16) 4 Section 2
The other possibility is that the photon is emitted from the second electron, corresponding to
e− time→ e− 1 ~p1 3
~p3
(17)
~p2
~p4 ¡ − − e2 e4
( A) 4 γ 2 which requires Vni = h ψe φ | V | ψe i . In this case, from the second electron’s point of view, the effect is felt before the source, the first electron, emitted the photon. The photon propagator in this case is called an advanced propagator. Obviously which diagram is advanced or retarded depends on what we call the source, but either way there are two intermediate states, one with a retarded and the other with an advanced propagator. To find an expression for these matrix elements, we insert our field operators:
( R) e 3 γ 1 e 3 3 γ 1 V = h ψe φ | V | ψe i = d x h ψe φ | ψe( x) φ( x) ψe ( x) | ψe i (18) ni 2 2 Z To evaluate this this, recall from from Lecture I-2, that the second quantized fields are
d3 p
Q Q Q 1 iQ p x † − i p x φ( xQ ) = a pe + a e (19) Z (2π) 3 2 ω p p p
with a similar form for the electron and that Q h φ γ | φ( x) | 0i = e − iQ pγ · x (20) and similarly for other matrix elements. The interaction V( x) is a product of three fields, and one can pair either of the electron fields in V( x) with either of the electron states in evaluating 3 γ 1
h ψe φ | V | ψe i , so we pick up a factor of 2 in the matrix element. We then find
Q Q
( R) 3 i ( Qp1 − p3 − pγ ) x 3 3
Q Q V = e d xe = e(2π) δ ( Qp − p3 − pγ) (21) ni Z 1
( A) The other matrix elements, Vni and those involving the final stare are similar, as you can check. Thus we have at first non-vanishing order 2
3 3 3 3 3 e
Q Q Q Q Q Q Tfi = d Qpγ(2π) δ ( p1 − p3 − pγ)(2π) δ ( p2 − p4 + pγ) (22) Z Ei − En These δ-functions tells us that 3-momentum is conserved in the local interactions between the photon and the electrons. Note that nothing tells us that energy is conserved; if it were, then En = Ei and this matrix element would blow up. This should not surprise you; the energy of intermediate states has always been different from the energy of the initial and final states in quantum mechanics – due to the uncertainty principle, energy can be not conserved for short times. To find a form for En, let us first denote the intermediate photon energy as Eγ, the incoming
electron energies E1 and E2 , and the outgoing electron energies E3 and E4. The momenta of the
Q Q Q
electrons are Qp1 ,p2 ,p3,p4 as above. By conservation of momentum, the photon momentum must
Q Q be Qpγ = p1 − p3. The photon energy is whatever it needs to be to put the photon on-shell: 0 =
2 2 2 Q pγ = Eγ − pQγ , so Eγ = | pγ | . That is
µ
Q Q Q Q pγ = ( Eγ ,pQ γ) = ( | p1 − p3 | ,p1 − p3 ) (23) For Eq. (22), we need the intermediate state energy, which is different for retarded and advanced cases. Lippmann-Schwinger equation 5
In the retarded case the first electron emits the photon and we look at the state before the photon hits the second electron, as shown in the figure in Eq. (15). In this case at the interme- diate time the first electron is already in its final state, with energy E3 and so the total interme- diate state energy is ( R) En = E3 + E2 + Eγ (24) So, dropping the 2π factors and the overall momentum conserving δ-functions for clarity (we will give a detailed derivation of these factors and their connection to scattering cross sections in the relativistic theory in next lecture), we find
e2 e2 e2 T( R) fi = ( R) = = (25) ( E1 + E2 ) − ( E3 + E2 + Eγ) ( E1 − E3) − Eγ Ei − En In the advanced case the second electron emits the photon and we look at the intermediate state before the photon hits the first electron, as in the diagram in Eq. (17). Then the energy is
( A) En = E4 + E1 + Eγ (26) and e2 e2 e2 T( A) fi = ( A) = = (27) ( E1 + E2 ) − ( E4 + E1 + Eγ) ( E2 − E4) − Eγ Ei − En
Finally, we have to add the advanced and retarded contributions, since they are both valid intermediate states. Overall energy conservation says E1 + E2 = E3 + E4, so E1 − E3 = E4 − E2 ≡ ∆E. So the sum is e2 e2 e2 e2 2e2 E T( R) T( A) γ fi + fi = R + A = + = 2 2 (28) ( ) ( ) ∆E − Eγ − ∆E − Eγ (∆E) − ( Eγ) Ei − En Ei − En To simplify this answer, let us define a 4-vector k µ by
µ µ µ
k ≡ p3 − p1 = (∆E,pQ γ) (29) Note, this is not the photon momentum in Eq. (23) since Eγ = | Qpγ | ∆E, or more simply, since 2 µ k 0. But k is a Lorentz 4-vector, since it comprises an energy and a 3-momentum. The norm of k µ is 2 2 2 k = (∆E) − ( Eγ) (30) This is convenient, since it lets us write the transition matrix simply as e2 Tfi = 2Eγ (31) k2
The 2Eγ is related to normalization which, along with the 2π and δ-function factors, will be properly accounted for in the relativistic treatment of the transfer matrix in the next chapter. T 1 1 The remarkable feature of fi is that it contains a Lorentz invariant factor of k 2 . This k 2 = 1 − is the Green’s function for a Lorentz invariant theory. If one of the electrons were at rest, we would sum the appropriate combination of momentum eigenstates which would amount to 1 Fourier transforming k 2 to reproduce the Coulomb potential, as in the previous lecture.
2.2 Feynman rules for OFPT Let us summarize some ingredients that went into the above example calculation: • All states are physical , that is, they are on-shell at all times.
• Matrix elements Vij will vanish unless 3-momentum is conserved at each vertex. • Energy is not conserved at each vertex. 6 Section 3
These are the Feynman rules for old-fashioned perturbation theory. As mentioned in the intro- duction, on-shell means that the state satisfies its free-field equations of motion. For example, a 2 2 2 2 2 2 scalar field satisfying ( + m ) φ = 0 would have p = m . It is called on-shell since Qp = E − m at fixed E and m is the equation for the surface of a sphere. So on-shell particles live on the shell of the sphere. Despite the fact that the intermediate states in OFPT are on-shell, we saw that it was µ 2 helpful to write the answer in terms of a Lorentz 4-vector k with k 0 representing the momentum of an unphysical, off-shell photon. We were led to k µ by combining two diagrams with different temporal orderings, which we called advanced and retarded. It would be nice if we could get k µ with just one diagram, where 4-momentum is conserved at vertices and so propaga- tors can be Lorentz invariant from the start. In fact we can! That’s what we’ll be doing for the rest of the course. As we will see, there is just one propagator in this approach, the Feynman propagator, which combines the advanced and retarded propagators into one in a beautifully efficient way. So we won’t have to keep track of what happens first. This new formalism will give us a much more cleanly organized framework to address the confusing infinities which plague quantum field theory calculations. Before finishing OFPT, as additional motivation and for its important historical relevance, we will heuristically review one such infinity.
3 Infinities Historically, one of the first confusions about the second-quantized photon field was that the Hamiltonian 3 d k † 1 H = ωk a ak + (32) Z (2π) 3 k 2 with ωk = kQ seemed to imply that the vacuum has infinite energy 1 d3k E = h 0| H | 0i = kQ = ∞ (33) 0 2 (2π) 3 Z
Fortunately, there is an easy way out of this paradoxical infinity: how do you measure the energy of the vacuum? You don’t! Only energy differences are measurable, and in these differ- ences the zero-point energy, the energy of the ground state, drops out. This is the basic idea behind renormalization – infinities can appear in intermediate calculations, but they must drop out of physical observables. This zero-point energy does have consequences, such as the Casimir effect (Lecture III-1) which comes from the difference in zero point energies in different size boxes, and the cosmological constant problem, which comes from the fact that energy gravitates. We will come to understand these two examples in detail later in the course, but it makes more sense to start with some less exotic physics. In 1930, Oppenheimer thought to use perturbation theory to compute the shift of the energy of the Hydrogen atom due to the photons. (R. Oppenheimer, Phys Rev . 35 461, 1930). He got infinity and concluded that quantum electrodynamics was wrong. In fact the result is not infinite but a finite calculable quantity known as the Lamb shift which agrees perfectly with data. However, it is instructive to understand Oppenheimer’s argument. 3.1 Oppenheimer and the Lamb shift Using old-fashioned perturbation theory we would calculate the energy shift using 2 |h ψn | Hint| ψm i| ∆En = h ψn | Hint| ψn i + (34) En − Em
mX n This is the standard formula from time-independent perturbation theory. The basic problem is that we have to sum over all possible intermediate states | ψm i , including ones that have nothing in particular to do with the system of interest (for example, free plane waves). It’s still true in field theory that there are only a finite number of states below any given energy level E, so that as E → ∞, 1 → 0. The catch is that there are an infinite number of states, and their phase E − En 3 space density goes like d3k ∼ E3 so that you get E → ∞ and perturbation theory breaks E − En down. This is exactly whatR Oppenheimer found. Infinities 7
First, take something where the calculation makes sense, such as a fixed non-dynamical back- ground field. Say there is an electric field in the zˆ direction. Then the potential energy is pro-
portional to the electric field Q Hint = eE · xQ = e | E | z (35) This interaction produces the linear Stark effect which is a straightforward application of time-independent perturbation theory in quantum mechanics. Our discussion of the Stark effect here will be limited to a quick demonstration that it is finite, and a representation of the result in terms of diagrams. Since an atom has no electric dipole moment, the first order correction is zero
h ψn | Hint| ψn i = 0 (36) At second order
2 |h ψ0 | Hint| ψm i| ∆E0 = = (37) E0 − Em
m>X0 ¡
The picture on the right is the corresponding Feynman diagram: the blobs represent the electric field which sources photons which interact with the electron, represented as the solid line on the bottom; the points where the photon meets the electron correspond to matrix elements of Hint; finally, the line between the two photon insertions is the electron propagator, the 1 factor E0 − Em in the second order expression for ∆E0. To show that ∆E0 is finite, we assume that E0 < 0 without loss of generality and that Em > E1 > E0 so that E0 is the ground state. Since ∆E0 < 0, by Eq. (37), we need to show that ∆E0 it is bounded from below. Using the completeness relation
1 = | ψm ih ψm | = | ψ0ih ψ0 | + | ψm ih ψm | (38) mX> 0 m>X0 we have
1 1 2 2 − ∆E0 6 h ψ0| Hint| ψm ih ψm | Hint| ψ0i = [ h ψ0| Hint| ψ0i − h ψ0| Hint| ψ0i ] (39) E1 − E0 E1 − E0 m>X0
The right hand side of this equation is a positive number, thus ∆E0 is bounded from below and above (by 0) and hence the energy correction to the ground state is finite. While it is not hard to calculate ∆E0 exactly for a given system, such as the Hydrogen atom, the only thing we want to observe here is that ∆E0 is finite. Now, instead of an external electric field, what would happen if this field were produced by the electron itself? Then we need to allow for the creation of photons by the electron and their annihilation back into the electron, which can be described with our second-quantized photon field. The starting Hamiltonian, for which we know the exact eigenstates, now has two parts atom photon H0 = H0 + H0 (40) with energy eigenstates given by electron wavefunctions associated with a set of photons, so
H0| ψn; { nk }i = En + nkωk | ψn; { nk }i (41) Xk
Where we allow for any number of excitations nk of the photons of any momenta kQ . At second order in perturbation theory, only one photon can be created and destroyed, but we have to integrate over this photon’s momentum. We are interested in the integration region where the photon has a very large momentum. By momentum conservation in old-fashioned per-
turbation theory, since the ground state only has support for small momentum, the exited state Q of the atom must have large momentum roughly backwards to that of the photon Qp ∼ − k . Thus
the excited state wavefunction will approach that of a free plane wave. The excited state energy Q is E ≈ | Qp | + k and so at large k, the integral will look like
Q Q 3 3 i ( k − Qp ) · x d p d k 3 e ∆E0 ∼ 3 3 d x (42) π π Q Z (2 ) Z (2 ) Z E0 − | Qp | + | k | 8 Section 3
3 Q Q After evaluating the x integral to get δ Qp − k and then the p integral, we find d3k 1 1 ∆E0 ∼ 3 = 2 kdk = ∞ (43) Z (2π) kQ 2π Z
This means that there should be an infinite shift in the energy levels of the Hydrogen atom. Oppenheimer also showed that if you take the difference between two levels, relevant for the shift in spectral lines, the result is also divergent. He concluded that “It appears improbable that the difficulties discussed in this work will be soluble without an adequate theory of the masses of the electron and proton; nor is it certain that such a theory will be possible on the basis of the special theory of relativity.” [Oppenheimer, p.477] What went wrong? In the Stark effect calculation we only had to sum over excited electron states, through m> 0 | ψm ih ψm | in Eq. (39), which was finite. For the Lamb shift calculation, the sum was alsoP over photon states which was divergent. It diverged because the phase space for photons, d3k is larger than the suppression 1 due to the energies of the intermediate kQ excited states. In terms of Feynman diagrams, the difference is that in the latter case we do not consider interactions with a fixed external field, but integrate over dynamical fields, corre- sponding to intermediate state photons. Since the photons relevant to the h ψ0| Hint | ψm ; 1 k i matrix element are the same as the photons relevant to the second, h ψm; 1 k | Hint| ψ0 i matrix ele- ment, the photons lines are the same and we can identify them. Thus the diagram contracts:
∆E0 = → (44)
¡ ¡ and the Stark effect diagram becomes a loop diagram for the Lamb shift. These pictures are just shorthand for the perturbation expansion, but they give us a nice mnemonic for the physics.
The loop means that there is an unknown momentum, kQ , which we have to integrate over. We’re only forcing momentum to be conserved overall, but it can be split between the atom and photon however it likes. There was actually nothing wrong with Oppenheimer’s calculation. He did get the answer that old-fashioned perturbation theory predicts. What he missed was that there are other infini- ties which eventually cancel this infinity (for example, the electron mass is infinite too, so in fact his conclusion was on the right track). This discussion was really just meant as a preview to demonstrate the complexities we’ll be up against. To sort out all these infinities, it will be really helpful, but not strictly necessary, to have a formalism which keeps the symmetries, in partic- ular Lorentz invariance, manifest along the way. Although Schwinger was able to tame the infinities using old-fashioned perturbation theory, in his own words, “Like the silicon chips of more recent years, the Feynman diagram was bringing computation to the masses.”