Second Quantization: Quantum Fields Bosons and Fermions Let Xj stand for the coordinate and spin subscript (if any) of the j-th particle, so that the vector of state jΨi of N particles has the form jΨi ≡ 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 permutational symmetry: Ψ(:::;Xi;:::Xj;:::) = ±Ψ(:::;Xj;:::Xi;:::); 8 Xi $ Xj: (1) The sign-plus particles are called bosons and the sign-minus particles are called fermions. Single-particle-based basis states Let fφs(X)g be an ONB of vectors of states for one particle. Then, the basis in the N-particle Hilbert space can be chosen as follows (fermions and bosons are labelled with the subscript F and B, respectively): φs (X1) φs (X2) : : : φs (XN ) 1 1 1 1 φ (X ) φ (X ) : : : φ (X ) s2 1 s2 2 s2 N ΨF (X1;X2;:::;XN ) = p ; (2) N! ······························ φsN (X1) φsN (X2) : : : φsN (XN ) X ΨB(X1;X2;:::;XN ) = C φs1 (X1)φs2 (X2) ··· φsN (XN ): (3) nontrv prm In the fermionic case, we have a compact determinant form, the so-called Slater 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: n ! n ! n ! ··· jCj2 = 1 2 3 : (4) N! Problem 1. Derive Eq. (4) [after reading the next section where the notion of occupation numbers is introduced]. Fock nomenclature and Fock space With the representations (2) and (3) for the basis states, it is quite clear that each basis state is exhaustively characterized by the set of N single-particle states, fφs1 ; φs2 ; : : : ; φsN g involved in the N-particle (anty-)symmetric basis state. For fermions, all the N states must be different.3 For bosons, there are no restrictions. Extremely powerful is the Fock nomenclature for the above-mentioned sets of single-particle states. In this approach, one globally enumerates all the single-particle states of a given set fφsg and then uses a string of natural numbers, (n(1); n(2); n(3);:::), such that 1 X n(s) = N; (5) s=1 to label all the N-particle basis states. The convention is that n(s) = 0, if (s) the state φs does not participate in the given basis state, while 1 ≤ n ≤ N (s) stands for the multiplicity of the state φs. Obviously, for fermions, n can be either 0 or 1. The next important step is to abandon the constraint (5) thus allowing the basis states with different numbers of particles.4 This way we arrive at the Fock space. Each vector of the Fock space can be represented as a linear combination of the Fock basis states. Furthermore, each Fock basis vector can be viewed as a direct product of single-mode (Fock) basis vectors: jn(1); n(2); n(3);:::i = jn(1)ijn(2)ijn(3)i · · · : (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). Elementary subsystems of which the quantum field consists are not the individual parti- cles. Rather, these are the single-particle modes. For each mode s, there is a complete set of orthonormal states fjn(s)ig, (s) (s) hn1 jn2 i = δ (s) (s) : (7) n1 n2 Recalling that n(s) = 0; 1 for fermions and n(s) = 0; 1; 2;::: for bosons, we see that, nomenclature-wise, the Fock space of fermions can be viewed as a set of two-level systems, while the Fock space of bosons corresponds to that of a set of quantum harmonic oscillators. The analogy between the quantum field of bosons and a set of independent quantum harmonic oscillators becomes essentially direct for a gas of noninteracting particles.5 Within the QFT picture, the number of particles is an observable rather than 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 for each mode: ^ X N = n^s; (8) s (s) (s) (s) n^sjn i = n jn i: (9) In this formalism, the occupation numbers of the mode s are the eigenvalues of the operatorn ^s. Ideal gas Consider now a noninteracting system. Choose fφs(X)g to be an ONB of single-particle energy eigenstates, with s corresponding single-particle en- ergy. Then, in the Fock representation, the Hamiltonian reads: ^ X H = sn^s: (10) s Note a qualitative difference between (10) and (8). While (10) requires that (i) the system be ideal and (ii) the set fφs(X)g be the set of single-particle energy 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 operatorsn ^s commute with each other, we conclude that the gas of ideal fermions is equivalent to a set of non-interacting two-level systems, while the ideal bosons are equivalent to a set of non-interacting harmonic oscillators. In both cases, the single-particle energies s 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: ^ 0 Y ^ 0 e−βH = e−βHs : (11) s Indeed, ^ 0 ^ ^ X ^ 0 ^ 0 H = H − µN = Hs; Hs = ~sn^s; ~s = s − µ. (12) s This way we reduce the problem of equilibrium statistics of ideal fermions or bosons to the equilibrium statistics of (respectively) a two-level system or a quantum harmonic oscillator. In the Fock representation, an explicit expression for the statistical oper- ator for the mode s is ^ 0 X (s) e−βHs = jn(s)i e−β~sn hn(s)j: (13) n(s) Up to a difference in the value of ~s, we have the same expression for each mode, so that the subscript s becomes redundant and we can write (^ρ ≡ ^ 0 e−βHs ) ρ^ = X jni e−βn~ hnj: (14) n The normalization constant7 z = Trρ ^ = X e−βn~ (15) n has a very deep statistical-mechanical meaning. This is the so-called partition function of the single mode. The partition function, Z, for the whole (ideal gas) system is then nothing but a product (here we restore the mode label) Y Z = zs: (16) s 6For clarity, we omit the normalization factor. 7The normalization rule isρ ^ ! ρ/z^ . 4 Formula (16) is central for the statistical mechanics of ideal Fermi and Bose gases.8 Problem 2. Calculate single-mode z as a function of (, µ, T ). The average occupation number of the mode, 1 1 n¯ = Trρ ^n^ = X n e−βn~ z z n is given by 1 n¯ = (Fermions); (17) eβ~ + 1 1 n¯ = (Bose): (18) eβ~ − 1 Problem 3. Derive (17) and (18). Hint. The following mathematical trick might be useful. @ X n eλn = X eλn: n @λ n In the limit β~ 1 (that is ~ T ), the difference between bosons and fermions disappears, since the contribution of n > 1 is exponentially suppressed: ρ^ ≈ j0ih0j + j1i e−/T~ h1j; n¯ ≈ e−/T~ (~ 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! −~s=T Problem 4. Show thatn ¯s = e 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, by simply calculating partial derivatives of the function T ln Z 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 with Maxwell'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 crucial to make sure that there is no nontrivial Jacobian associated with going from P R 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 sense to choose the most convenient ones. Namely, a finite-size box of rectangular shape with periodic boundary conditions. For bosons, there is yet another classical limit. It takes place at ~ T and corresponds to the classical limit of corresponding harmonic oscillator. In terms of the QFT perspective, this limit corresponds to the classical-field behavior 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: N N N ^(1) X ^(1) ^(2) 1 X X ^(2) F = f (Xj); F = f (Xj;Xk);::: j=1 2 j=1 k=1(6=j) (20) Second quantization is the formalism allowing one to represent the operators F^(1); F^(2);:::; etc.