Second Quantization: Quantum Fields

Bosons and Fermions

LetXj stand for the coordinate and spin subscript (if any) of the j-th particle,so that the vector of state |Ψ〉 of N particles has the form

|Ψ〉 ≡ Ψ(X1, X2, . . . , XN).

The postulate of indistinguishability states that the Hilbert space of par-ticles of the same kind1 is restricted to the states featuring permutationalsymmetry:

Ψ(. . . , Xi, . . . Xj, . . .) = ±Ψ(. . . , Xj, . . . Xi, . . .), ∀Xi ↔ Xj. (1)

The sign-plus particles are called bosons and the sign-minus particles arecalled fermions.

Single-particle-based basis states

Let {φs(X)} be an ONB of vectors of states for one particle. Then, the basisin the N -particle Hilbert space can be chosen as follows (fermions and bosonsare labelled with the subscript F and B, respectively):

ΨF (X1, X2, . . . , XN) =1√N !

∣∣∣∣∣∣∣∣∣φs1(X1) φs1(X2) . . . φs1(XN)φs2(X1) φs2(X2) . . . φs2(XN)· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·φsN (X1) φsN (X2) . . . φsN (XN)

∣∣∣∣∣∣∣∣∣ , (2)

ΨB(X1, X2, . . . , XN) = C∑

nontrv prm

φs1(X1)φs2(X2) · · ·φsN (XN). (3)

In the fermionic case, we have a compact determinant form, the so-calledSlater determinant. For bosons, the summation is over all non-trivial2 per-mutations of pairs of coordinates. The bosonic normalization constant C can

1Referred to as identical particles.2The permutation is trivial if it does not change the form of the term. This happens if

and only if the the corresponding functions φ are the same.

1

be conveniently expressed in terms of the occupation numbers of correspond-ing single-particle states:

|C|2 =n1!n2!n3! · · ·

N !. (4)

Problem 1. Derive Eq. (4) [after reading the next section where the notionof occupation numbers is introduced].

Fock nomenclature and Fock space

With the representations (2) and (3) for the basis states, it is quite clear thateach basis state is exhaustively characterized by the set of N single-particlestates, {φs1 , φs2 , . . . , φsN} involved in the N -particle (anty-)symmetric basisstate. For fermions, all the N states must be different.3 For bosons, thereare no restrictions. Extremely powerful is the Fock nomenclature for theabove-mentioned sets of single-particle states. In this approach, one globallyenumerates all the single-particle states of a given set {φs} and then uses astring of natural numbers, (n(1), n(2), n(3), . . .), such that

∞∑s=1

n(s) = N, (5)

to label all the N -particle basis states. The convention is that n(s) = 0, ifthe state φs does not participate in the given basis state, while 1 ≤ n(s) ≤ Nstands for the multiplicity of the state φs. Obviously, for fermions, n(s) canbe either 0 or 1.

The next important step is to abandon the constraint (5) thus allowingthe basis states with different numbers of particles.4 This way we arrive atthe Fock space. Each vector of the Fock space can be represented as a linearcombination of the Fock basis states. Furthermore, each Fock basis vectorcan be viewed as a direct product of single-mode (Fock) basis vectors:

|n(1), n(2), n(3), . . .〉 = |n(1)〉|n(2)〉|n(3)〉 · · · . (6)

3Otherwise, the Slater determinant is identically equal to zero.4By definition, the basis states with different total numbers of particles are orthogonal

to each other.

2

Factorization (6) is central for casting the theory of many-body identical-particle systems in the form of quantum field theory (QFT). Elementarysubsystems of which the quantum field consists are not the individual parti-cles. Rather, these are the single-particle modes. For each mode s, there isa complete set of orthonormal states {|n(s)〉},

〈n(s)1 |n

(s)2 〉 = δ

n(s)1 n

(s)2. (7)

Recalling that n(s) = 0, 1 for fermions and n(s) = 0, 1, 2, . . . for bosons, we seethat, nomenclature-wise, the Fock space of fermions can be viewed as a setof two-level systems, while the Fock space of bosons corresponds to that of aset of quantum harmonic oscillators. The analogy between the quantum fieldof bosons and a set of independent quantum harmonic oscillators becomesessentially direct for a gas of noninteracting particles.5

Within the QFT picture, the number of particles is an observable ratherthan a fixed external parameter. The operator of the total number of par-ticles, N , is constructed as a sum of operators of occupation of numbers foreach mode:

N =∑s

ns, (8)

ns|n(s)〉 = n(s)|n(s)〉. (9)

In this formalism, the occupation numbers of the mode s are the eigenvaluesof the operator ns.

Ideal gas

Consider now a noninteracting system. Choose {φs(X)} to be an ONB ofsingle-particle energy eigenstates, with εs corresponding single-particle en-ergy. Then, in the Fock representation, the Hamiltonian reads:

H =∑s

εsns. (10)

Note a qualitative difference between (10) and (8). While (10) requires that(i) the system be ideal and (ii) the set {φs(X)} be the set of single-particleenergy eigenstates, none of the two requirements is relevant to (8).

5No matter whether the system of bosons is uniform or placed in an external potential.

3

Given that, by construction, different operators ns commute with eachother, we conclude that the gas of ideal fermions is equivalent to a set ofnon-interacting two-level systems, while the ideal bosons are equivalent to aset of non-interacting harmonic oscillators. In both cases, the single-particleenergies εs play the role of inter-level separations.

Gibbs distribution. In particular, the above observation means that the(grand canonical) Gibbs statistical operator6 factorizes into a product of(grand canonical) Gibbs statistical operators for each single-particle mode:

e−βH′=∏s

e−βH′s . (11)

Indeed,

H ′ = H − µN =∑s

H ′s, H ′s = εsns, εs = εs − µ. (12)

This way we reduce the problem of equilibrium statistics of ideal fermions orbosons to the equilibrium statistics of (respectively) a two-level system or aquantum harmonic oscillator.

In the Fock representation, an explicit expression for the statistical oper-ator for the mode s is

e−βH′s =

∑n(s)

|n(s)〉 e−βεsn(s)〈n(s)|. (13)

Up to a difference in the value of εs, we have the same expression for eachmode, so that the subscript s becomes redundant and we can write (ρ ≡e−βH

′s)

ρ =∑n

|n〉 e−βεn〈n|. (14)

The normalization constant7

z = Tr ρ =∑n

e−βεn (15)

has a very deep statistical-mechanical meaning. This is the so-called partitionfunction of the single mode. The partition function, Z, for the whole (idealgas) system is then nothing but a product (here we restore the mode label)

Z =∏s

zs. (16)

6For clarity, we omit the normalization factor.7The normalization rule is ρ→ ρ/z.

4

Formula (16) is central for the statistical mechanics of ideal Fermi and Bosegases.8

Problem 2. Calculate single-mode z as a function of (ε, µ, T ).

The average occupation number of the mode,

n =1

zTr ρ n =

1

z

∑n

n e−βεn

is given by

n =1

eβε + 1(Fermions), (17)

n =1

eβε − 1(Bose). (18)

Problem 3. Derive (17) and (18). Hint. The following mathematical trickmight be useful. ∑

n

n eλn =∂

∂λ

∑n

eλn.

In the limit βε � 1 (that is ε � T ), the difference between bosonsand fermions disappears, since the contribution of n > 1 is exponentiallysuppressed:

ρ ≈ |0〉〈0| + |1〉 e−ε/T 〈1|, n ≈ e−ε/T (ε� T ). (19)

This “ultra quantum” (in terms of the statistics of occupation numbers)regime actually corresponds to the regime of Boltzmann gas of classical par-ticles!

Problem 4. Show that ns = e−εs/T corresponds to the Maxwell distribu-tion of classical particles, once s is understood as the momentum eigenvalue,

8In statistical mechanics, one extracts all the thermodynamic quantities from Z, bysimply calculating partial derivatives of the function T lnZ with respect to temperature,or/and chemical potential, or/and volume.

5

s ≡ p. Note that it is not sufficient to simply identify the exponential withMaxwell’s exponential. It is equally important to demonstrate that the sum-mation over s can be replaced with the integration dp; that is, it is crucialto make sure that there is no nontrivial Jacobian associated with going from∑s(. . .) to

∫(. . .)dp. Hint. Since, in a macroscopic system, the answer is in-

dependent of the system’s shape and the boundary conditions, it makes senseto choose the most convenient ones. Namely, a finite-size box of rectangularshape with periodic boundary conditions.

For bosons, there is yet another classical limit. It takes place at ε � Tand corresponds to the classical limit of corresponding harmonic oscillator.In terms of the QFT perspective, this limit corresponds to the classical-fieldbehavior of corresponding mode.

Second quantized form of operators I: Statement of the problem

Consistently with the distinguishability postulate, all the relevant opera-tors/observables should be permutational symmetric. The class of permu-tational symmetric operators/observables further splits into single-particle,F (1), two-particle, F (2), etc., subclasses:

F (1) =N∑j=1

f (1)(Xj), F (2) =1

2

N∑j=1

N∑k=1(6=j)

f (2)(Xj, Xk), . . .

(20)Second quantization is the formalism allowing one to represent the operatorsF (1), F (2), . . . , etc. directly within the Fock space, without explicitly resort-ing to their N -body X-dependence.

Second quantized form of operators II: bosons

By explicit calculation, we find that the only non-zero off-diagonal9 matrixelements for the operator F (1) are the following ones:

〈n(s), (n(r)− 1) | F (1) | (n(s)− 1), n(r) 〉 = f (1)sr

√n(s)n(r), (21)

9That is, connecting two different states.

6

f (1)sr =

∫φ∗s(X) f (1)(X)φr(X) dX. (22)

In the two different Fock states of Eq. (21), we explicitly show only those oc-cupation numbers that are different between the two states. The integrationover dX is understood as integration over dr and summation over the spinsubscript (if any).

For the diagonal matrix elements of the operator F (1), we find

〈 F (1) 〉 =∑s

f (1)ss n

(s). (23)

A crucial observation about Eqs. (21)–(23) is that (i) the explicit dependenceon N disappears and (ii) the X-dependence enters only the single-particlematrix element (22).

Problem 5. Derive Eqs. (21)–(23).

Creation and annihilation operators. The annihilation operator, bs, (of themode s) is defined as

bs |n(s) 〉 =√n(s) |n(s)− 1 〉. (24)

〈n(s)− 1 | bs |n(s) 〉 =√n(s) . (25)

The creation operator, b†s, (of the mode s) is defined as a Hermitianconjugate to bs. We thus have

〈n(s) | b†s |n(s)− 1 〉 = 〈n(s)− 1 | bs |n(s) 〉∗ =√n(s), (26)

b†s |n(s) 〉 =√n(s) + 1 |n(s)+ 1〉. (27)

From (24) and (27) we then readily see that

b†s bs = n(s), (28)

bsb†s = n(s) + 1. (29)

With (24), (27), (28), and (29), we find the commutation relations10

[bs, br] = 0, [b†s, b†r] = 0, (30)

10It is very instructive to compare those relations to the Poisson brackets of complexcanonical variables in classical mechanics, replacing Hermitian conjugation with complexconjugation and replacing the commutator with the Poisson bracket.

7

[bs, b†r] = δsr. (31)

Now we can straightforwardly check that, in terms of the operators {bs}and {b†s} acting directly in the Fock space, we have

F (1) =∑s,r

f (1)sr b

†s br. (32)

The few-body interactions have similar structure. In particular,

F (2) =1

2

∑s,r,q,t

f(2)sr;qt b

†s b†r bq bt, (33)

where

f(2)sr;qt =

∫φ∗s(X1)φ∗r(X2) f (2)(X1, X2)φq(X2)φt(X1) dX1 dX2. (34)

As a by-product of establishing relations (32), we can readily establish alaw of transformation of creation and annihilation operators when changingthe single-particle ONB generating the Fock basis. Indeed, suppose we gofrom one single-particle basis to the other by the unitary transformation

φs =∑r

Usrφr, φs =∑r

U∗rsφr, (35)

where U is the unitary matrix of the transformation. Then, the matrixelements (22) get transformed with the matrix U , and for the form (32) tostay invariant,11 the creation/annihilation operators have to be transformedas

b†s =∑r

Usrˆb†

r,ˆb†

s =∑r

U∗rsb†r, (36)

bs =∑r

U∗srˆbr,

ˆbs =∑r

Ursbr. (37)

Field operator. The field operator, ψ(X), is introduced as follows:

ψ(X) =∑s

bs φs(X). (38)

11Note that Eq. (32) was derived for any basis.

8

It is also useful to introduce

ψ†(X) ≡ [ψ(X)]+ =∑s

b†s φ∗s(X). (39)

A prominent feature of the operator ψ(X) is that it is invariant with re-spect to the transformation (35)–(37). Also worth mentioning are the com-mutation relations—following directly from the definition (38):

[ψ(X1), ψ(X2)] = [ψ†(X1), ψ†(X2)] = 0, (40)

[ψ(X1), ψ†(X2)] =∑s

φs(X1)φ∗s(X2) = δ(r2 − r1) δσ1σ2 , (41)

where we recall that X ≡ (r, σ), with σ the spin subscript (for spinelessparticles, δσ1σ2 should be omitted).

With the field operator, the expressions for F (1), F (2), . . . , etc. can becast into a compact, basis-invariant (quantum-field-theoretical) form:

F (1) =∫ψ†(X)f (1)(X)ψ(X) dX, (42)

F (2) =1

2

∫ψ†(X1)ψ†(X2)f (2)(X1, X2)ψ(X2)ψ(X1) dX1dX2. (43)

The particle (number) density operator, n(r), has the form

n(r) =∑σ

ψ†(X)ψ(X). (44)

Indeed, in the original N -particle coordinate representation, the operatorn(r) has the F (1)-type form with a delta-functional f (1):

n(r) =N∑j=1

δ(r− rj). (45)

The rest follows from (42).Integrating n(r) over r, we are supposed to get the total number of par-

ticles. Hence, the operator of the total number of particles is

N =∫ψ†(X)ψ(X) dX. (46)

9

Second quantized form of operators III: fermions

In the case of fermions, we proceed similarly to what we have done for bosons.Here, however, we have to deal with one (annoying) circumstance. Namely,the fact that the sign of the basis wave function (2) depends on the orderin which single-particle functions are listed in the determinant. Swappingany two of them, φs ↔ φr, changes the global sign of the wave function(2). Hence, to unambiguously fix the sign of the matrix elements betweendifferent Fock states, we have to fix the signs of the functions (2). To thisend, we rely on the above-introduced global enumeration of all the modes,s = 1, 2, 3, . . ., and require that corresponding single=particle functions inthe determinant (2) be ordered is such a way that

s1 < s2 < s3 < . . . < sN . (47)

With this enumeration, the following integer-valued function—defined for agiven Fock basis state—proves relevant

Θ(s, r) = Θ(r, s) =

{ ∑max(s,r)−1q=min(s,r)+1 n

(q), if |s− r| > 1,

0, otherwise.(48)

The meaning of Θ(s, r) is very simple. It is the sum of the occupationnumbers of all the modes (if any) between the modes s and r, presuming theordering of modes by the above-mentioned enumeration. An analog of (21)then is

〈 1s, 0r | F (1) | 0s, 1r 〉 = f (1)sr (−1)Θ(s, r), (49)

where 0s stands for n(s) = 0 and 1s stands for n(s) = 1. It is not surprisingthat, apart from the sign factor, the result (49) corresponds to (21), withthe occupation numbers not exceeding unity. To verify the sign factor in(49), observe that, if the two Fock basis functions between which the ma-trix element is calculated were related to each other by simply replacingφs ↔ φr in the Slater determinant (2), then no sign would occur.12 In ac-cordance with our ordering convention, the state | 0s, 1r 〉 can be obtainedfrom the state | 1s, 0r 〉 in two steps: (i) replacing φs → φr and (ii) assign-ing the sign (−1)Θ(s, r) to compensate for the wrong13 position of the row[φr(X1), φr(X2), . . . , φr(XN)] in the determinant (2).

12Because each term in the bra state would have the same sign as its counterpart in theket state.

13Speaking generally.

10

For the diagonal element, we get the same expression (23).

Creation and annihilation operators for fermions. The annihilation and cre-ation operators (of the mode s), as and a†s, are defined as operators having(only) the following non-zero matrix elements

〈0s|as|1s〉 = 〈1s|a†s|0s〉 = (−1)γs , (50)

γs =

{ ∑s−1r=1 n

(r), if s > 1,0, if s = 1.

(51)

There is a qualitative difference with the bosonic case, where the annihila-tion and creation operators are defined for each single-particle mode indi-vidually. This is not true for fermions. Now the sign of the matrix elementdepends on the occupation numbers of the other modes, and the action ofcreation/annihilation operators of different modes is not independent!

The crucial properties of creation/annihilation operators are as follows.

〈 1s, 0r | a†s ar | 0s, 1r 〉 = (−1)Θ(s, r), (52)

〈 1s, 0r | ar a†s | 0s, 1r 〉 = −(−1)Θ(s, r), (53)

anda†s as = n(s), (54)

as a†s = 1− n(s). (55)

Combining (52) and (54) with (49) and (23) [it is readily seen that (23) worksfor Fermions as well], we arrive at the fermionic analog of (32):

F (1) =∑s,r

f (1)sr a

†s ar. (56)

Combining (52) and (53), we find the following anti-commutation relation

a†s ar + ara†s = 0 (s 6= r). (57)

From (54) and (55) we see that

a†s as + asa†s = 1, (58)

11

so that (57) and (58) can be combined into a single relation

a†s ar + ara†s = δsr. (59)

Similarly, starting from the definition (50), we can establish yet another setof anti-commutation relations

as ar + aras = 0, a†s a†r + a†ra

†s = 0, (60)

implying, in particular,

as as = 0, a†s a†s = 0. (61)

Problem 6. Starting from the definition (50), derive (52) through (55) andalso (60).

It is not only Eq. (56) that exactly reproduces its bosons counterpart. Thesame situation takes place for all the formulas derived for bosons in terms ofcreation and annihilation operators, as well as in terms of the field operators

ψ(X) =∑s

as φs(X), ψ†(X) =∑s

a†s φ∗s(X), (62)

with one obvious exception for the commutation properties of the field op-erators. Given the anti-commutative properties of fermionic creation andannihilation operators, we readily see that fermionic analogs of bosonic com-mutators of field operators are the following anti-commutators (below {A, B}stands for AB + BA):

{ψ(X1), ψ(X2)} = {ψ†(X1), ψ†(X2)} = 0, (63)

{ψ(X1), ψ†(X2)} =∑s

φs(X1)φ∗s(X2) = δ(r2 − r1) δσ1σ2 . (64)

The only subtlety is that now—because of the anti-commutativity—one shouldbe extremely careful with the order of operators in fermionic analogs ofEqs. (33) and (43). Being aware of this circumstance, we made sure that theorder of bosonic operators in Eqs. (33) and (43) is such that their fermionic

12

analogs are obtained simply by replacing b→ a, no minus sign emerging.

Problem 7. Based on (anti-)commutation relations for creation/annihilationoperators of bosons and fermions, prove Eqs. (40), (41), (63), and (64).

The transformation (35) of the single-particle ONB implies corresponding(the very same as for bosons) transformation of the creation/annihilationoperators:

a†s =∑r

Usr ˆa†r, ˆa

†s =

∑r

U∗rsa†r, (65)

as =∑r

U∗sr ˆar, ˆas =∑r

Ursar. (66)

Problem 8. Make sure that the transformations (36)–(37) and (65)–(66)preserve corresponding (anti-)commutation properties of the creation/annihilationoperators.

Axiomatic quantum field theory

We introduce quantum field operators

ψ(X) =∑s

as φs(X), ψ†(X) ≡ [ψ(X)]+ =∑s

a†s φ∗s(X), (67)

such that {φs(X)} is a certain single-particle ONB, and the (annihilationand creation) operators {as} and {a†s ≡ (as)

+} are postulated to obey eitherbosonic or fermionic (anti-)commutation rules:

[as, ar] = 0, [as, a†r] = δsr (bosons), (68)

{as, ar} = 0, {as, a†r} = δsr (fermions). (69)

The axioms imply corresponding (anti-)commutational properties of the fieldoperators. The representational invariance of the axiomatic quantum fieldtheory is established by the observation that the transformation (36)–(37),on one hand, preserves the bosonic and fermionic (anti-)commutational prop-erties, and, on the other hand, preserves the structure of the field operators,

13

if accompanied by corresponding transformation (35) of the single-particleONB.

The Fock space is defined as a space in which the creation/annihilationoperators act. A special axiom postulates the existence of the vacuum state,14

|vac〉, as the (unique) state featuring the following property:

as |vac〉 = 0, ∀s. (70)

The Fock ONB then is defined as the set of the following vectors obtainedby acting with the creation operators on the vacuum state:

|n(1), n(2), n(3), . . .〉 =

(a†1)n(1)

√n(1)!

(a†2)n(2)

√n(2)!

(a†3)n(3)

√n(3)!

· · ·

|vac〉. (71)

The integers n(s) = 0, 1, 2, . . . are called occupation numbers of the mode s.A special theorem (Problem 9) establishes the fact that the states (71) areorthonormalized.

For fermions, the anticommutation axiom (69) implies a†sa†s = 0, so that

the occupation numbers can be either 0 or 1 (Pauli principle), simplifying(71) to

|n(1), n(2), n(3), . . .〉 =[(a†1)n

(1)

(a†2)n(2)

(a†3)n(3) · · ·

]|vac〉 (fermions). (72)

With fermions, one can further simplify notation by mentioning only non-zero occupation numbers:

|1s1 , 1s2 , . . . , 1sN 〉 = a†s1 a†s2· · · a†sN |vac〉, s1 < s2 < . . . < sN . (73)

For bosons, it is very instructive to observe that the axioms imply thatthe Fock space and the algebra of creation/annihilation operators reduce tothose of independent harmonic oscillators.

Problem 9. Prove that (i) each state (71) is normalized to unity and (ii)any two different states (71) are orthogonal. Hint. It is helpful to derive andthen use the following formula

as(a†s)m = m(a†s)

m−1 + (a†s)mas (bosons). (74)

14Not to be confused with the zero vector.

14

The relations (24) through (29) and (50) through (55) are derived from ax-ioms.

Problem 10. Perform the above-mentioned derivations.

15