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Quantization of Relativistic Free Fields

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Quantization of Relativistic Free Fields Life can only be understood backwards, but it must be lived forwards. S. Kierkegaard (1813-1855) 7 Quantization of Relativistic Free Fields In Chapter 2 we have shown that quantum mechanics of the N-body Schr¨odinger equation can be replaced by a Schr¨odinger field theory, where the fields are oper- ators satisfying canonical equal-time commutation rules. They were given in Sec- tions 2.8 and 2.10 for bosons and fermions, respectively. A great advantage of this reformulation of Schr¨odinger theory was that the field quantization leads automati- cally to symmetric or antisymmetric N-body wave functions, which in Schr¨odinger theory must be imposed from the outside in order to explain atomic spectra and Bose-Einstein condensation. Here we shall generalize the procedure to relativistic particles by quantizing the free relativistic fields of the previous section following the above general rules. For each field φ(x, t) in a classical Lagrangian L(t), we define a canonical field momentum as the functional derivative (2.156) δL(t) π(x, t)= px(t)= . (7.1) δφ˙(x, t) The fields are now turned into field operators by imposing the canonical equal-time commutation rules (2.143): [φ(x, t),φ(x′, t)] = 0, (7.2) [π(x, t), π(x′, t)] = 0, (7.3) [π(x, t),φ(x′, t)] = iδ(3)(x x′). (7.4) − − For fermions, one postulates corresponding anticommutation rules. In these equations we have omitted the hat on top of the field operators, and we shall do so from now on, for brevity, since most field expressions will contain quantized fields. In the few cases which do not deal with field operators, or where it is not obvious that the fields are classical objects, we shall explicitly state this. We shall also use natural units in which c =1 andh ¯ = 1, except in some cases where CGS-units are helpful. Before implementing the commutation rules explicitly, it is useful to note that all free Lagrangians constructed in the last chapter are local, in the sense that they are given by volume integrals over a Lagrangian density L(t)= d3x (x, t), (7.5) Z L 474 7.1 Scalar Fields 475 where (x, t) is an ordinary function of the fields and their first spacetime deriva- tives. TheL functional derivative (7.1) may therefore be calculated as a partial deriva- tive of the density (x)= (x, t): L L ∂ (x) π(x)= L . (7.6) ∂φ˙(x) Similarly, the Euler-Lagrange equations may be written down using only partial derivatives ∂ (x) ∂ (x) ∂µ L = L . (7.7) ∂[∂µφ(x)] ∂φ(x) Here and in the sequel we employ a four-vector notation for all spacetime objects such as x =(x0, x), where x0 = ct coincides with the time t in natural units. The locality is an important property of all present-day quantum field theories of elementary particles. The spacetime derivatives appear usually only in quadratic terms. Since a derivative measures the difference of the field between neighbor- ing spacetime points, the field described by a local Lagrangian propagates through spacetime via a “nearest-neighbor hopping”. In field-theoretic models of solid-state systems, such couplings are useful lowest approximations to more complicated short- range interactions. In field theories of elementary particles, the locality is an essential ingredient which seems to be present in all fundamental theories. All nonlocal effects arise from higher-order perturbation expansions of local theories. 7.1 Scalar Fields We begin by quantizing the field of a spinless particle. This is a scalar field φ(x) that can be real or complex. 7.1.1 Real Case The action of the real field was given in Eq. (4.167). As we shall see in Subsec. 7.1.6, and even better in Subsec. 8.11.1, the quanta of the real field are neutral spinless particles. For the canonical quantization we must use the real-field Lagrangian density (4.169) in which only first derivatives of the field occur: 1 (x)= [∂φ(x)]2 M 2φ2(x) . (7.8) L 2{ − } The associated Euler-Lagrange equation is the Klein-Gordon equation (4.170): ( ∂2 M 2)φ(x)=0. (7.9) − − 7.1.2 Field Quantization According to the rule (7.6), the canonical momentum of the field is ∂ (x) π(x)= L = ∂0φ(x)= φ˙(x). (7.10) ∂φ˙(x) 476 7 Quantization of Relativistic Free Fields As in the case of non-relativistic fields, the quantization rules (7.2)–(7.4) will even- tually render a multiparticle Hilbert space. To find it we expand the field operator φ(x, t) in normalized plane-wave solutions (4.180) of the field equation (7.9): ∗ † φ(x)= fp(x)ap + fp(x)ap . (7.11) p X h i Observe that in contrast to a corresponding expansion (2.209) of the nonrelativis- tic Schr¨odinger field, the right-hand side contains operators for the creation and annihilation of particles. This is an automatic consequence of the fact that the quantizations of φ(x) and π(x) lead to Hermitian field operators, and this is natural for quantum fields in the relativistic setting, which necessarily allow for the cre- ∗ ation and annihilation of particles. Inserting for fp(x) and fp(x) the explicit wave functions (4.180), the expansion (7.11) reads 1 φ(x)= (e−ipxa + eipxa† ). (7.12) 0 p p p √2p V X Since the zeroth component of the four-momentum pµ in the exponent is on the mass shell, we should write px more explicitly as px = ωpt px. However, the notation − px is shorter, and there is little danger of confusion. If there is such a danger, we 0 shall explicitly state the off- or on-shell property p = ωp. The expansions (7.11) and (7.12) are complete in the Hilbert space of free par- † ticles. They may be inverted for ap and ap with the help of the orthonormality relations (4.177), which lead to the scalar products † ∗ a =(fp,φ) , ap = (f ,φ). (7.13) p t − p More explicitly, these can be written as ap 0 1 = e±ip x0 d3x e∓ipx iπ(x)+ p0φ(x) , (7.14) ( a† ) √2Vp0 ± p Z h i where we have inserted π(x)= φ˙(x). Using the canonical field commutation rules (7.2)–(7.4) between φ(x) and π(x), we find for the coefficients of the plane-wave expansion (7.12) the commutation relations † † [ap, ap′ ] = ap, ap′ =0, † h i ap, ap′ = δp,p′ , (7.15) h i making them creation and annihilation operators of particles of momentum p on a vacuum state 0 defined by ap 0 = 0. Conversely, reinserting (7.15) into (7.12), we | i | i verify the original local field commutation rules (7.2)–(7.4). 7.1 Scalar Fields 477 In an infinite volume one often uses, in (7.11), the wave functions (4.181) with continuous momenta, and works with a plane-wave decomposition in the form of a Fourier integral d3p 1 φ(x)= e−ipxa + eipxa† , (7.16) (2π)3 2p0 p p Z with the identification [recall (2.217)] 0 † 0 † ap = 2p V ap, ap = 2p V ap. (7.17) q q The inverse of the expansion is now 0 ap ±ip x0 3 ∓ipx 0 † = e d x e iπ(x)+ p φ(x) . (7.18) ( ap ) ± Z h i This can of course be written in the same form as in (7.13), using the orthogonality relations (4.182) for continuous-momentum wave functions (4.181): † ∗ a = (fp,φ) , ap = (f ,φ) . (7.19) p t − p t In the expansion (7.11), the commutators (7.15) are valid after the replacement 0 -(3) ′ 0 3 (3) ′ δp p′ 2p δ (p p )=2p (2πh¯) δ (p p ), (7.20) , → − − which is a consequence of (1.190) and (1.196). In the following we shall always prefer to work with a finite volume and sums over p. If we want to find the large-volume limit of any formula, we can always go over to the continuum limit by replacing the sums, as in (2.257), by phase space integrals: 3 V d p 3 , (7.21) p → (2πh¯) X Z 3 (2πh¯) (3) ′ δp p′ δ (p p ). (7.22) , → V − After the replacement (7.17), the volume V disappears in all physical quantities. A single-particle state of a fixed momentum p is created by p = a† 0 . (7.23) | i p| i Its wave function is obtained from the matrix elements of the field 1 −ipx 0 φ(x) p = e = fp(x), h | | i √2Vp0 1 p φ(x) 0 = eipx = f ∗(x). (7.24) h | | i √2Vp0 p 478 7 Quantization of Relativistic Free Fields As in the nonrelativistic case, the second-quantized Hilbert space is obtained by applying the particle creation operators a† to the vacuum state 0 , defined by p | i a† 0 = 0. This produces basis states (2.219): p| i S,A † np † np np np ...np = (ˆa ) 1 (ˆa ) k 0 . (7.25) | 1 2 k i N p1 ··· pk | i Multiparticle wave functions are obtained by matrix elements of the type (2.221). The single-particle states are p =a† 0 . (7.26) | i p| i They satisfy the orthogonality relation (p′ p)=2p0 δ-(3)(p′ p). (7.27) | − Their wave functions are −ipx (0 φ(x) p) = e =fp(x), | | (p φ(x) 0) = eipx =f∗(x). (7.28) | | p Energy of Free Neutral Scalar Particles On account of the local structure (7.5) of the Lagrangian, also the Hamiltonian can be written as a volume integral H = d3x (x).
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