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Fock Space Formalism, the Net Operator (52) Acts As

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Fock Space Formalism, the Net Operator (52) Acts As Free Fields, Harmonic Oscillators, and Identical Bosons A free quantum field and its canonical conjugate are equivalent to a family of harmonic oscillators (one oscillator for each plane wave), which is in turn equivalent to a quantum theory of free identical bosons. In this note, I will show how all of this works for the relativistic scalar field' ^(x) and its conjugateπ ^(x). And then I will turn around and show that a quantum theory of any kind of identical bosons is equivalent to a family of oscillators. (Harmonic for the free particles, non-harmonic if the particles interact with each other.) Moreover, for the non-relativistic particles, the oscillator family is in turn equivalent to a non-relativistic quantum field theory. In this note we shall work in the Schr¨odingerpicture of Quantum Mechanics because it's more convenient for dealing with the eigenstates and the eigenvalues. Consequently, all operators | including the quantum fields such as' ^(x) | are time-independent. From Relativistic Fields to Harmonic Oscillators Let us start with the relativistic scalar field' ^(x) and its conjugateπ ^(x); they obey the canonical commutation relations [' ^(x); '^(x0)] = 0; [^π(x); π^(x0)] = 0; [' ^(x); π^(x0)] = iδ(3)(x − x0) (1) and are governed by the Hamiltonian Z ^ 3 1 2 1 2 1 2 2 H = d x 2 π^ (x) + 2 (r'^(x)) + 2 m '^ (x) : (2) We want to expand the fields into plane-wave modes' ^k andπ ^k, and to avoid technical difficulties with the oscillators and their eigenstates, we want discrete modes. Therefore, we replace the infinite x space with a finite but very large box of size L × L × L, and impose periodic boundary conditions |' ^(x + L; y; z) =' ^(x; y + L; z) =' ^(x; y; z + L) =' ^(x; y; z), etc., etc. For large L, the specific boundary conditions are unimportant, so I have chosen 1 the periodic conditions since they give us particularly simple plane-wave modes 2π (x) = L−3=2eikx where k ; k ; k = × an integer: (3) k x y z L Expanding the quantum fields into such modes, we get Z X −3=2 ikx 3 −3=2 −ikx '^(x) = L e × '^k; '^k = d x L e × '^(x); k Z (4) X −3=2 ikx 3 −3=2 −ikx π^(x) = L e × π^k; π^k = d x L e × π^(x): k A note on hermiticity: The classical fields '(x) and π(x) are real (i.e., their values are real numbers), so the corresponding quantum fields are hermitian,' ^y(x) =' ^(x) andπ ^y(x) = π^(x). However, the mode operators' ^k andπ ^k are not hermitian; instead, eqs. (4) give us y y '^k =' ^−k andπ ^k =π ^−k . The commutation relations between the mode operators follow from eqs. (1), namely [' ^k; '^k0 ] = 0; [^πk; π^k0 ] = 0; [' ^k; π^k0 ] = i δk;−k0 : (5) The first two relations here are obvious, but the third needs a bit of algebra: Z Z 3 3 0 −3 −ikx −ik0x0 0 [' ^k; π^k0 ] = d x d x L e e × [' ^(x); π^(x )] Z Z 0 0 = d3x d3x0 L−3e−ikxe−ik x × iδ(3)(x − x0) (6) Z 0 = i L−3 d3x e−ix(k+k ) box = i δk;−k0 : Equivalently, y y y y [' ^k; π^k0 ] = [' ^k; π^k0 ] = i δk;k0 ; [' ^k; π^k0 ] = [' ^k; π^k0 ] = i δk+k0;0 : (7) 2 Now let's express the Hamiltonian (2) in terms of the modes. For the first term, we have Z Z Z 3 2 3 y 3 X X −3 −ikx +ik0x y d x π^ (x) = d x π^ (x)^π(x) = d x L e e × π^kπ^k0 k k0 0 1 Z X y −3 3 ix(k0−k) 0 (8) = π^kπ^k0 × @L d x e = δk;k A 0 k;k box X y = π^kπ^k : k Similarly, the last term becomes Z 3 2 X y d x '^ (x) = '^k'^k ; (9) k while in the second term X −3=2 ikx X −3=2 −ikx y r'^(x) = L e × ik '^k = L e × −ik '^k ; (10) k k hence Z 3 2 X 2 y d x (r'^(x)) = k '^k'^k : (11) k Altogether, the Hamiltonian (2) becomes ^ X 1 y 1 2 2 y H = 2 π^kπ^k + 2 (k + m )' ^k'^k : (12) k Clearly, this Hamiltonian describes a bunch of harmonic oscillators with frequencies !k = p k2 + m2 (in theh ¯ = c = 1 units). But since the mode operators are not hermitian, converting them into creation and annihilation operators takes a little more work then usual: 3 We define 1 a^k = p !k'^k + i π^k ; 2!k and consequently y 1 y y a^k = p !k'^k − i π^k ; 2!k (13) 1 1 y y a^−k = p !−k'^−k + i π^−k = p !k'^k + i π^k ; 2!−k 2!k y 1 y y 1 a^−k = p !−k'^−k − i π^−k = p !k'^k − i π^k : 2!−k 2!k y y Note thata ^k 6=a ^−k anda ^−k 6=a ^k; instead, we have independent creation and annihilation y operatorsa ^k anda ^k for every mode k. The commutations relations between these operators are y y y [^ak; a^k0 ] = 0; [^ak; a^k0 ] = 0; [^ak; a^k0 ] = δk;k0 : (14) Indeed, 1 0 0 [^a ; a^ 0 ] = p !! [' ^ ; '^ 0 ] + i! [^π ; '^ 0 ] + i! [' ^ ; π^ 0 ] − [^π ; π^ 0 ] k k 4!!0 k k k k k k k k 1 0 = p 0 + i! × −iδk+k0;0 + i! × +iδk+k0;0 + 0 4!!0 (15) !0 − ! = δk+k0;0 × p 4!!0 = 0 because !0 = ! when k + k0 = 0: y y Similarly, [^ak; a^k0 ] = 0. Finally, y 1 0 y 0 y y y [^a ; a^ 0 ] = p !! [' ^ ; '^ 0 ] + i! [^π ; '^ 0 ] − i! [' ^ ; π^ 0 ] + [^π ; π^ 0 ] k k 4!!0 k k k k k k k k 1 0 = p 0 + i! × −iδk;k0 − i! × +iδk;k0 + 0 4!!0 (16) ! + !0 = δk;k0 × p 4!!0 0 0 = δk;k0 because ! = ! when k = k: To re-obtain the field mode operators' ^k andπ ^k from the creation and annihilation y operators, let us combine the first and the last equations (13) for thea ^k anda ^−k . Adding 4 and subtracting those equations, we find r y p y 2 a^k +a ^−k = 2!k × '^k ; −ia^k + ia^−k = × π^k : (17) !k Consequently, ! !2'^y '^ = k × (^ay +a ^ )(^a +a ^y ) k k 2 k −k k −k ! = k × a^y a^ +a ^y a^y +a ^ a^ +a ^ a^y ; 2 k k k −k −k k −k −k ! (18) π^y π^ = k × (ia^y − ia^ )(−ia^ + ia^y ) k k 2 k −k k −k ! = k × a^y a^ − a^y a^y − a^ a^ +a ^ a^y ; 2 k k k −k −k k −k −k hence 2 y y y y ! '^k'^k +π ^kπ^k = !k × a^ka^k +a ^−ka^−k (19) y y = !k × a^ka^k +a ^−ka^−k + 1 : Altogether, the Hamiltonian (12) becomes ^ X 1 y y X y 1 H = 2 !k × a^ka^k +a ^−ka^−k + 1 = !k a^ka^k + 2 : (20) k k In light of the commutation relations (14), this Hamiltonian clearly describes an infinite family of harmonic oscillators, one oscillator for each plane-wave mode k. Now consider the eigenvalues and the eigenstates of the multi-oscillator Hamiltonian (20). 1 A single harmonic oscillator has eigenvalues En = !(n + 2 ) where n = 0; 1; 2; 3;:::. For the y multi-oscillator system at hand, eachn ^k =a ^ka^k commutes with all the othern ^k0 , so we may diagonalize them all at the same time. This gives us eigenstates O X jfn for all kgi = jn i of energy E = ! (n + 1 ): (21) k k fnkg k k 2 k k where each nk is an integer ≥ 0. Moreover, all combinations of the nk are allowed because y thea ^k anda ^k operators can change a particular nk ! nk ± 1 without affecting any other 5 y 0 nk0 . (This follows from [^ak; n^k0 ] = 0 and [^ak; n^k0 ] = 0 for k 6= k.) Thus, the Hilbert space of the multi-oscillator system | and hence of the free quantum field theory | is a direct product of Hilbert spaces for each oscillator, O H(QFT) = H(harmonic oscillator for mode k): (22) k From the Multi-Oscillator System to Identical Bosons A constant term in the Hamiltonian of a quantum system does not affect its dynamics in any way, it simply shifts energies of all states by the same constant amount. So to simplify our analysis of the multi-oscillator system in particle terms, let's subtract the infinite zero- P 1 point energy E0 = k 2 !k from the Hamiltonian (20), thus ^ ^ X y H ! H − E0 = !ka^ka^k : (23) k I'll come back to the zero-point energy, but right now let's focus on other issues. In the multi-oscillator Hilbert space (22) each occupation number nk is independent from P all others. However, states of finite energy must have finite N = k nk, so let us re-organize ^ P the Hilbert space into eigenspaces of the N = k n^k operator, 1 O M H(QFT) = H(mode k) = HN ; (24) k N=0 and consider what do those eigenspaces look like for different N.
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