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221B Lecture Notes
Quantum Field Theory (a.k.a. Second Quantization)
1 Quantum Field Theory
Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content, yet applicable to wider varieties of systems than the conventional formulation of quantum mechanics.
1.1 Limitations of Multi-body Schr¨odinger Wave Func-
tion
We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odinger wave function, and can never change the number of arguments. Similarly, it is useful to consider elementary excitations in multi-body systems, such as phonons, and they can be created or annihilated by putting energy into the system. Finally even the number of matter particles changes in relativistic elementary paticle physics by pair creations, pair annihilations etc. The limitaions of multi-body Schr¨odinger wave function mentioned here
call for a better formalism. The quantum field theory is designed specifically for that.
1.2 Aim of the Quantum Field Theory
Quantum Field Theory is sometimes called “2nd quantization.” This is a very bad misnomer because of the reason I will explain later. But nonetheless, you are likely to come across this name, and you need to know it. The aim of the quantum field theory is to come up with a formalism which is completely equivalent to multi-body Schr¨odinger equations but just better: it allows you to consider a variable number of particles all within the same framework and can even describe the change in the number of particles. It also gives totally symmetric or anti-symmetric multi-body wave function au- tomatically, not as a consequence of symmetrization or anti-symmetrization done “by hand.” Moreover, once we go to relativistic systems, quantum field theory is the only known formalism to describe quantum mechanics consis- tent with Lorentz invariance and causality. It also allows a systematic way of organizing perturbation theory in terms of “Feynman diagrams,” a graphical representation of each term in perturbation theory. Even though the contents of multi-body quantum mechanics and quantum field theory are exactly the same in systems where the number of particles does not change, it turns out that the field theory is a far better formalism for many purposes and is widely used in condensed matter physics, elementary particle physics, quantum optics, and in some cases also in atomic/molecular physics and nuclear physics. It is particularly suited to multi-body problems. In systems where the number of particles changes, or where one needs super- position of states with different number of particles (sounds odd, but we will see examples later), quantum field theory is crucial. But it is clear that the formalism will confuse you (at least) once be- cause of too-close resemblance but conceptual difference from the ordinary Schr¨odinger equation. I will try to make it as clear as possible below.
1.3 Particle-Wave Duality
It is interesting to note that the way quantum mechanics and quantum field theory work is a sort of the opposite. In quantum mechanics, you start with classical particle Hamiltonian mechanics, with no concept of wave or interfer- ence. After quantizing it, we introduce Schr¨odinger wave function and there
This classical field equation can be derived from the action
S =
∫ dtd~x φ∗(~x, t)
( i
∂t
2 μ
) φ(~x, t). (5)
Note that the Lagrangian for a field is given by an integral over space, and in this case S =
∫ dtL(t) =
∫ dtd~x L(~x, t) (6)
where L(~x, t) is called Lagrangian density. By taking the variation of the action with respect to φ∗^ → φ∗^ + δφ∗, we find
δS =
∫ dtd~x δφ∗(~x, t)
( i
∂t
2 μ
) φ(~x, t), (7)
and requiring that the action must be stationary, we recover the field equation Eq. (2). You may worry that the action Eq. (5) is not real; but it can be made real by adding surface terms
S =
∫ dtd~x
( i 2
(φ∗^ φ˙ − φ˙∗φ) −
2 μ
∇~φ∗^ ∇~φ
)
. (8)
But for simplicity we use the previous form Eq. (5) for the rest of the note. Because the classical field theory does not need to be linear unlike quan- tum mechanical Schr¨odinger equation, we can add a non-linear term (higher than quadratic term) in the action, for instance,
S =
∫ d~xdt
( iφ∗^ φ˙ + φ∗^
2 μ
φ −
γφ∗^2 φ^2
)
. (9)
The parameter γ has a dimension of M −^1 L. Following the same variation, we find a non-linear field equation
( i
∂t
2 μ
− γφ∗φ
) φ(~x, t) = 0. (10)
This equation is non-linear and we cannot solve this equation exactly in general any more.
3 Quick Review of Quantization Procedure
Let us briefly review how we quantize a particle mechanics system. Given an action on the phase space (pi, qi),
S =
∫ dt L =
∫ dt (
∑
i
pi q˙i − H(pi, qj )), (11)
the variational principle gives the Hamilton equations of motion
δS δpi
= ˙qi −
∂H
∂pi
δS δqi
= − p˙i −
∂H
∂qi
You can again add a surface term and make it look symmetric between p and q as L =
∑ i
1 2 (pi^ q˙i^ −^ qi^ p˙i)^ −^ H(p, q). To quantize this system, we first identify the canonically conjugate momentum of the coordinate qi by
pi =
∂L
∂ q˙i
Then we set up the canonical commutation relation
[pi, pj ] = [qi, qj ] = 0, [pi, qj ] = −i¯hδij. (15)
This defines the quantum theory. The index i runs over all degrees of free- dom in the system. The quantum mechanical state is given by a ket |ψ(t)〉, which admits a “coordinate representation” 〈q|ψ〉 = ψ(q, t). The dynamical evolution of the system is given by the Schr¨odinger equation
i¯h
∂t
|ψ(t)〉 = H(pi, qj )|ψ(t)〉. (16)
It is also useful to recall the commutation relation between creation and annihilation operator of harmonic oscillators
[ai, a† j ] = δij , [a, a] = [a†, a†] = 0. (17)
Here, I assumed there are many harmonic oscillators labeled by the subscript i or j. The Hilbert space is constructed from the ground state | 0 〉 which satisfies ai| 0 〉 = 0 (18)
This defines the quantum theory of the Schr¨odinger field.^1 The canonical commutation relation Eq. (21) is very similar to the case of the harmonic oscillator. We now go back to the normalization ψ(~x) = φ(~x)/¯h^1 /^2 , and we find [ψ(~x), ψ†(~y)] = δ(~x − ~y), [ψ, ψ] = [ψ†, ψ†] = 0, (22)
which resembles the case of harmonic oscillator even better. Now you see that the use of ψ(~x) was more convenient (we are not afraid of having ¯h in the Lagrangian any more because we are discussing a quantum theory anyway). We now regard ψ(~x) as annihilation operator and ψ†(~x) creation operator of a particle at position ~x. The Hamiltonian of the system from the action Eq. (9) is read off in the same way as in Eq. (11):
H =
∫ d~x
( φ†^
2 μ
φ +
γ(φ†φ)^2
)
∫ d~x
( ψ†^
−¯h^2 ∆ 2 m
ψ +
λψ†^2 ψ^2
)
. (23)
Here I recovered m = ¯hμ, and introduced λ = ¯h^2 γ to save space (and ink). Note that H| 0 〉 = 0 (24)
because ψ(~x) in the Hamiltonian annihilates directly the vacuum. A general state must satisfy the Schr¨odinger equation in the same way as in particle mechanics
ih ¯
∂t
|Ψ(t)〉 = H|Ψ(t)〉 (25)
It is useful for later purpose to calculate the commutator [H, ψ†(~x)].
[H, ψ†(~x)] =
∫ d~y
[ ψ†(~y)
−¯h^2 ∆~y 2 m
ψ(~y) +
λψ†^2 (~y)ψ(~y)^2 , ψ†(~x)
]
( −¯h^2 ∆~x 2 m
ψ†(~x) +
λψ†^2 (~x)2ψ(~x)
)
. (26)
(^1) The commutation relation is given in the Schr¨odinger picture, where the opera-
tors do not evolve in time while the states do. When switching to Heisenberg picture, we have to specify that the canonical commutation relation holds only at equal times: [π(~x, t), φ(~y, t)] = [iφ†(~x, t), φ(~y, t)] = −i¯hδ(~x − ~y). Commutation relation of operators at unequal times would depend on the dynamics of the system.
4.2 Fock Space
The canonical commutation relation Eq. (21) allows us to construct the Hilbert space following the experience of the harmonic oscillator. The par- ticular Hilbert space we construct below is called Fock space. We define the “vacuum” | 0 〉 which is annihilated by the annihilation op- erator ψ(~x)| 0 〉 = 0, (27)
and construct the Fock space by
|~x 1 , ~x 2 , · · · , ~xN 〉 = ψ†(~x 1 )ψ†(~x 2 ) · · · ψ†(~xN )| 0 〉. (28)
You can use (ψ(~x 1 ))n^1 as well, but we will not need it for the later discussions. What is the physical meaning of the Fock space we have constructed? It turns out that the state |~x 1 , ~x 2 , · · · , ~xn〉 in Eq. (28) has a very simple meaning: it is an n-particle state of identical bosons in the position eigenstate at ~x 1 , · · · , ~xn. We will verify this interpretation below explicitly.
4.3 One-particle State
Let us first study the one-particle state
|~x〉 = ψ†(~x)| 0 〉. (29)
The first quantity we calculate is its norm. Here we go:
〈~x|~y〉 = 〈 0 |ψ(~x)ψ†(~y)| 0 〉 = 〈 0 |[ψ(~x), ψ†(~y)]| 0 〉 = 〈 0 |δ(~x − ~y)| 0 〉 = δ(~x − ~y). (30)
Therefore, this state is normalized in the same way as the one-particle posi- tion eigenstate. Recall that a general one-particle state in the usual formulation of quan- tum mechanics is given by a linear superposition of position eigenstates as
|ψ〉 =
(∫ |~x〉d~x〈~x|
) |ψ〉 =
∫ |~x〉d~xψ(~x), (31)
The first quantity we calculate is again its norm. Here we go:
〈~x 1 , ~x 2 |~y 1 , ~y 2 〉 =
〈 0 |ψ(~x 2 )ψ(~x 1 )ψ†(~y 1 )ψ†(~y 2 )| 0 〉
=
〈 0 |ψ(~x 2 )([ψ(~x 1 ), ψ†(~y 1 )] + ψ†(~y 1 )ψ(~x 1 ))ψ†(~y 2 )| 0 〉
=
〈 0 |ψ(~x 2 )δ(~x 1 − ~y 1 )ψ†(~y 2 ) + ψ(~x 2 )ψ†(~y 1 )ψ(~x 1 ))ψ†(~y 2 )| 0 〉
=
〈 0 |δ(~x 1 − ~y 1 )[ψ(~x 2 ), ψ†(~y 2 )] + [ψ(~x 2 ), ψ†(~y 1 )][ψ(~x 1 )), ψ†(~y 2 )]| 0 〉
=
(δ(~x 1 − ~y 1 )δ(~x 2 − ~y 2 ) + δ(~x 1 − ~y 2 )δ(~x 2 − ~y 1 )). (37)
This normalization suggests that we are dealing with a two-particle state of identical particles, because the norm is non-vanishing when ~x 1 = ~y 1 and ~x 2 = ~y 2 , but also when ~x 1 = ~y 2 and ~x 2 = ~y 1 , i.e., two particles interchanged. A general two-particle state is given by
|Ψ(t)〉 =
∫ d~x 1 d~x 2 Ψ(~x 1 , ~x 2 , t)|~x 1 , ~x 2 〉 =
∫ d~x 1 d~x 2 Ψ(~x 1 , ~x 2 )ψ†(~x 1 )ψ†(~x 2 )| 0 〉. (38) Because [ψ†(~x 1 ), ψ†(~x 2 )] = 0, the integration over ~x 1 and ~x 2 is symmetric under the interchange ~x 1 ↔ ~x 2 , and hence Ψ(~x 1 , ~x 2 , t) = Ψ(~x 2 , ~x 1 , t). The symmetry under the exchange suggests that we are dealing with identical bosons. The fact that we see two identical particles becomes clearer by again working out the time evolution of a general state. Using the Schr¨odinger equation Eq. (25) with the Hamiltonian Eq. (23) and the state Eq. (36), the l.h.s of the Schr¨odinger equation is
i¯h
∂t
|Ψ(t)〉 =
∫ d~x 1 d~x 2
( i¯h
∂t
Ψ(~x 1 , ~x 2 , t)
) |~x 1 , ~x 2 〉. (39)
On the other hand, the r.h.s of the Schr¨odinger equation is
√ 2 H|Ψ(t)〉 =
∫ d~x 1 d~x 2 Ψ(~x 1 , ~x 2 , t)Hψ†(~x 1 )ψ†(~x 2 )| 0 〉
=
∫ d~x 1 d~x 2 Ψ(~x 1 , ~x 2 , t)([H, ψ†(~x 1 )] + ψ†(~x 1 )H)ψ†(~x 2 )| 0 〉
∫ d~x 1 d~x 2
( −¯h^2 ∆~x 1 2 m
ψ†(~x 1 ) +
λψ†^2 (~x 1 )2ψ(~x 1 )
) Ψ(~x 1 , ~x 2 , t)ψ†(~x 2 )| 0 〉
∫ d~x 1 d~x 2 Ψ(~x 1 , ~x 2 , t)ψ†(~x 1 )[H, ψ†(~x 2 )]| 0 〉
∫ d~x
( ψ†(~x 1 )
−¯h^2 ∆~x 1 2 m
) Ψ(~x 1 , ~x 2 , t)ψ†(~x 2 )| 0 〉
∫ d~x 1 d~x 2 ψ†(~x 1 )
( −¯h^2 ∆~x 2 2 m
ψ†(~x 2 ) +
λψ†^2 (~x 2 )2ψ(~x 2 )
) Ψ(~x 1 , ~x 2 , t)| 0 〉
∫ d~x 1 d~x 2
( −h¯^2 ∆~x 1 2 m
−¯h^2 ∆~x 2 2 m
) Ψ(~x 1 , ~x 2 , t)|~x 1 , ~x 2 〉.
(40)
Comparing Eqs. (39,40), we find
i¯h
∂t
Ψ(~x 1 , ~x 2 , t) =
( −¯h^2 ∆~x 1 2 m
−¯h^2 ∆~x 2 2 m
) Ψ(~x 1 , ~x 2 , t) (41)
which is nothing but a Schr¨odinger equation for two-particle wave func- tion Ψ(~x, t), with a delta function potential as an interaction between them. Therefore, the Fock space with two creation operators correctly describes the two-particle quantum mechanics. If we want a general interaction potential between them, the action Eq. (9) must be modified to
S =
∫ dt
[∫ d~x
( ψ∗i¯h ψ˙ + ψ∗^
¯h^2 ∆ 2 m
ψ
) −
∫ d~xd~yψ∗(~x)ψ∗(~y)V (~x − ~y)ψ(~y)ψ(~x)
] . (42) The corresponding Hamiltonian is
H =
∫ d~xψ∗^
−h¯^2 ∆ 2 m
ψ +
∫ d~xd~yψ∗(~x)ψ∗(~y)V (~x − ~y)ψ(~y)ψ(~x). (43)
You can follow exactly the same steps as above and derive the equation
i¯h
∂t
Ψ(~x 1 , ~x 2 , t) =
( −¯h^2 ∆~x 1 2 m
−h¯^2 ∆~x 2 2 m
) Ψ(~x 1 , ~x 2 , t). (44)
In summary, the quantized Schr¨odinger field correctly describes the two in- teracting identical bosons appropriately.
4.5 Three and More Particles
Following the same analysis as two-particle case, we can consider the three- particle state
|~x 1 , ~x 2 , ~x 3 〉 =
ψ†(~x 1 )ψ†(~x 2 )ψ†(~x 3 )| 0 〉, (45)
and also
[N, ψ†(~x)] =
∫ d~y[ψ†(~y)ψ(~y), ψ†(~x)] =
∫ d~yψ†(~y)δ(~y − ~x) = ψ†(~x). (52)
It is useful to rewrite this commutator as
N ψ†(~x) = ψ†(~x)(N + 1). (53)
In other words, every time you change the order of the number operator N and the creation operator ψ†, N increases by one. Note also
N | 0 〉 = 0, (54)
because the annihilation operator in N acts directly on the vacuum. Then the eigenvalue of N on n-body state can be read off quite easily.
N |~x 1 , · · · , ~xn〉 =
n!
N ψ†(~x 1 ) · · · ψ†(~xn)| 0 〉
n!
ψ†(~x 1 )(N + 1) · · · ψ†(~xn)| 0 〉 = · · · =
n!
ψ†(~x 1 ) · · · ψ†(~xn)(N + n)| 0 〉
= n|~x 1 , · · · , ~xn〉. (55)
We see that the operator N indeed picks up the number of the particles in a given state as its eigenvalue.
4.7 Momentum Space
It may be more intuitive to consider creation and annihilation operators in the momentum space, because the Hamiltonian is “diagonal” in the momentum space in the absence of the interaction potential. Suppose that the whole system is in a cubic box of volume V = L^3 with a periodic boundary condition. This choice of the boundary condition is often called the “box normalization.” Then the field operator can be expanded in the Fourier series
ψ(~x) =
L^3 /^2
∑
~p
a(~p)ei~p·~x/¯h, (56)
where
~p =
2 π¯h L
(nx, ny, nz ) (57)
with integer nx,y,z. The inverse transform is therefore
a(~p) =
L^3 /^2
∫ d~xψ(~x)e−i~p·~x/¯h. (58)
It is straightforward to calculate the commutation relations among the cre- ation and annihilation operators in the momentum space:
[a(~p), a†(~q)] =
L^3
∫ d~xd~y [ψ(~x)e−i~p·~x/¯h, ψ†(~y)ei~q·~y/¯h]
=
L^3
∫ d~xd~y δ(~x − ~y)e−i~p·~x/¯hei~q·~y/¯h
=
L^3
∫ d~xe−i(~p−~q)·~x/¯h = δ~p,~q. (59)
This is nothing but the commutation relation for harmonic oscillators. Ob- viously, [a(~p), a(~q)] = [a†(~p), a†(~q)] = 0. (60) It is instructive to rewrite the Hamiltonian in the momentum space. The free part of the Hamiltonian is
H 0 =
∫ d~xψ∗^
−h¯^2 ∆ 2 m
ψ
∫ d~x
L^3
∑
p ~
∑
~q
a†(~p)e−i~p·~x/¯h^
−¯h^2 ∆ 2 m
a(~q)ei~q·~x/¯h
∫ d~x
L^3
∑
p ~
∑
~q
a†(~p)e−i~p·~x/¯h^
~q^2 2 m
a(~q)ei~q·~x/¯h
L^3
∑
~p
∑
~q
a†(~p)
~q^2 2 m
a(~q)L^3 δ~p,~q
∑
~p
~p^2 2 m
a†(~p)a(~p). (61)
The free Hamltonian simply countes the number of particles in a given mo- mentum state a†(~p)a(~p) and assigns the energy ~p^2 / 2 m accordingly. The in-
is
H =
∫ d~xψ∗
( −¯h^2 ∆ 2 m
Ze^2 |~x|
) ψ +
∫ d~xd~y
ψ∗(~x)ψ∗(~y)
e^2 |~x − ~y|
ψ(~y)ψ(~x).
(64) where the second term in the parantheses is the Coulomb pontential due to the nucleus, while the last term is the Coulomb repulsion among electrons. In this case, a more convenient expansion of the field operator would be
ψ(~x) =
∑
nlm
anlmunlm(~x), (65)
where anlm is the annihilation operator and the c-number unlm(~x) is the complete basis for expanding the field operator with. Then the state with certain states filled can be written down as
a† 1 sa† 2 p,m=1| 0 〉 etc. (66)
But this Hamiltonian gives bosons instead of fermions. How to obtain fermions out of quantized Schr¨odinger field is the issue in the next section.
5 Fermions
We have seen that the quantized Schr¨odinger field gives multi-body states of identical bosons. But we also need fermions to describe electrons, protons, etc. How do we do that? The trick is to go back to the commutation relations Eq. (22). Instead of them, we can use anti-commutation relations
{ψ(~x), ψ†(~y)} = δ(~x − ~y), {ψ, ψ} = {ψ†, ψ†} = 0. (67)
The notation is {A, B} = AB +BA instead of [A, B] = AB −BA. This small change flips the statistics completely and the same Hamiltonian Eq. (43) describes a system of identical fermions. One noteworthy point is that ψ(~x)†^2 = 12 {ψ(~x)†, ψ(~x)†} = 0. What this means is that one cannot create two particles at the same position, an ex- pression of Pauli’s exclusion principle for fermions.
Here are a few useful identities. Similarly to the identity of commutators [A, BC] = [A, B]C + B[A, C], we find
[A, BC] = ABC − BCA = ABC + BAC − BAC − BCA = {A, B}C − B{A, C} (68)
Similarly,
[AB, C] = ABC − CAB = ABC + ACB − ACB − CAB = A{B, C} − {A, C}B. (69)
Again it is useful to calculate the commutator [H, ψ†(~x)] for later pur- poses.
[H, ψ†(~x)]
=
[∫ d~y
( ψ†(~y)
−¯h^2 ∆~y 2 m
ψ(~y) +
∫ d~z
ψ†(~y)ψ†(~z)V (~y − ~z)ψ(~z)ψ(~y)
) , ψ†(~x)
]
∫ d~yψ†(~y)
−¯h^2 ∆~y 2 m
{ψ(~y), ψ†(~x)}
∫ d~yd~z
ψ†(~y)ψ†(~z)V (~y − ~z)
[ ψ(~z)ψ(~y), ψ†(~x)
]
−¯h^2 ∆~x 2 m
ψ†(~x) +
∫ d~zψ†(~x)ψ†(~z)V (~x − ~z)ψ(~z). (70)
Consider a two-particle state
∫ d~x 1 d~x 2 Ψ(~x 1 , ~x 2 )ψ†(~x 1 )ψ†(~x 2 )| 0 〉. (71)
Because {ψ†(~x 1 ), ψ†(~x 2 )} = 0, or more explicitly
ψ†(~x 1 )ψ†(~x 2 ) = −ψ†(~x 2 )ψ†(~x 1 ), (72)
the c-number function Ψ is hence anti-symmetric under the exchange of two positions Ψ(~x 1 , ~x 2 ) = −Ψ(~x 2 , ~x 1 ). (73)
Such a state indeed describes identical fermions.
