Heisenberg–Lagrange–Hamilton quantum field mechanicsmechanics

5.2.5 Heisenberg–Lagrange–Hamilton quantum field mechanics

Finally, we are ready to quantize classical field formalism, and arrive at a quantum field mechanics – at least for the scalar field φ(x, t). If we were dealing with the case in which φ(x, t) represented the displacement of a one-dimensional stretched string, quantization would be straightforward. We would take the classical Hamiltonian (5.113) and promote the mode coordi­

nates Q_r and their conjugate momenta P_r to operators satisfying commutation relations of the form (5.85). The rest of the analysis would be exactly as in equations (5.86) to (5.89), except that the number of modes N is infinite. But in the case of the general scalar field, we do not want to impose the boundary conditions φ(0, t) = φ(e, t) = 0, which led to the mode expansion (5.34). It is then not so clear how to proceed.

Fortunately, the Lagrange-Hamilton field formalism does indicate the way forward, which is one good reason for developing it in the first place. (Another is that it is very well suited to the analysis of symmetries, a crucial aspect of gauge theories – see chapter 7.) In the previous section we introduced the

‘coordinate-like’ field φ(x, t) and (via the Lagrangian) the ‘momentum-like’

field π(x, t). To pass to the quantized version of the field theory, we mimic the procedure followed in the discrete case and promote both the quantities φ andπ to operators φˆ and ˆπ, in the Heisenberg picture. As usual, the distinctive feature of quantum theory is the non-commutativity of certain basic quantities in the theory – for example, the fundamental commutator (ħ = 1)

[ˆq_r(t), pˆ_s(t)] = iδ_rs (5.114) of the discrete case. Thus we expect that the operators φˆ and ˆπ will obey some commutation relation which is a continuum generalization of (5.114).

The commutator will be of the form [φˆ(x, t), πˆ(y, t)], since – recalling fig­

ure 5.5 – the discrete index r or s becomes the continuous variable x or y; we

also note that (5.114) is between operators at equal times. The continuum generalization of the δ_rs symbol is the Dirac δ function, δ(x − y), with the properties

∫ _∞

−∞ δ(x) dx = 1 (5.115)

∫ _∞

δ(x−y)f(x) dx = f(y) (5.116)

−∞

for all reasonable functions f (see appendix E). Thus the fundamental com­

mutator of quantum field theory is taken to be

[φˆ(x, t), πˆ(y, t)] = iδ(x− y) (5.117) in the one-dimensional case, with obvious generalization to the three-dimen­

sional case via the symbol δ³(x − y). Remembering that we have set ħ = 1, it is straightforward to check that the dimensions are consistent on both sides. Variables φˆ and ˆπ obeying such a commutation relation are said to be ‘conjugate’ to each other.

What about the commutator of two φˆ’s or two ˆπ’s? In the discrete case, two different ˆq’s (in the Heisenberg picture) will commute at equal times, [ˆq_r(t), qˆ_s(t)] = 0, and so will two different ˆp’s. We therefore expect to supple­

ment (5.117) with

[φˆ(x, t), φˆ(y, t)] = [ˆπ(x, t), πˆ(y, t)] = 0. (5.118) Let us now proceed to explore the effect of these fundamental commutator assumptions, for the case of the Lagrangian density which yielded the wave equation via the Euler–Lagrange equations, namely

( )2 ( )2

1 ∂φˆ 1 ∂φˆ

Lˆρ = ρ − ρc² . (5.119)

2 ∂t 2 ∂x

If we remove ρ, and set c = 1, we obtain

( )2 ( )2

1 ∂φˆ 1 ∂φˆ

L ˆ= − (5.120)

2 ∂t 2 ∂x

for which the Euler–Lagrangian equation yields the field equation

∂² φˆ ∂² φ ˆ

− = 0. (5.121)

∂t² ∂x²

We can think of (5.121) as a highly simplified (spin-0, one-dimensional) ver­

sion of the wave equation satisfied by the electromagnetic potentials. We may guess, then, that the associated quanta are massless, as we shall soon confirm.

139 5.2. The quantumfield: (ii) Lagrange–Hamilton formulation

The Lagrangian density (5.120) is our prototype quantum field Lagrangian (one often slips into leaving out the word ‘density’). Applying the quantized version of (5.95) we then have

∂Lˆ

It is not immediately clear how to find the eigenvalues and eigenstates of the operator Hˆ . However, it is exactly at this point that all our preliminary work on normalmodescomes into its own. If we can write the Hamiltonian as some kind of sum over independent oscillators – i.e. modes – we shall know how to proceed. For the classical string with fixed end points which was considered in section 5.1, the mode expansion was simply a Fourier expansion. In the present case, we want to allow the field to extend throughout all of space, without the periodicity imposed by fixed-end boundary conditions. In that case, the Fourier series is replaced by a Fourier integral, and standing waves are replaced by travelling waves. For the classical field obeying the wave equation (5.30) there are plane-wave solutions

ikx−iωt

φ(x, t) ∝ e (5.125)

where (c = 1)

ω = k (5.126)

which is just the dispersion relation of light in vacuo. The general field may be Fourier expanded in terms of these solutions:

∫ _∞ are purely conventional, and determine the normalization of the expansion coefficients a, a ^∗ and ˆa, ˆa^† later; in turn, the latter enter into the definition, and normalization, of the states – see (5.143)). Similarly, the ‘momentum field’ π = φ ˙ is expanded as

We quantize these mode expressions by promoting φ → φˆ, π → πˆ and assum­

ing the commutator (5.117). Thus we write

∫ _∞ φˆ = d

√ k

[ˆa(k)e^ikx−iωt + ˆa ^†(k)e^−ikx+iωt] (5.129)

−∞ 2π 2ω

and similarly for πˆ. The commutator (5.117) nowdeterminesthe commutators of the modeoperators aˆ and ˆa^†:

[ˆa(k), aˆ^†(k ^′ )] = 2πδ(k −k ^′ )

(5.130) [ˆa(k), aˆ(k ^′ )] = [ˆa^†(k), aˆ^†(k ^′ )] = 0

as shown in problem 5.6. These are the desired continuum analogues of the discrete oscillator commutation relations

[ˆa_r, aˆ^† _s] = δ_rs

(5.131) [ˆa_r, aˆ_s] = [ˆa^† _r, aˆ^†_s] = 0.

The precise factor in front of the δ-function in (5.130) depends on the normal­

ization choice made in the expansion of φˆ, (5.129). Problem 5.6 also shows that the commutation relations (5.130) lead to (5.118) as expected.

The form of the ˆa, ˆa^† commutation relations (5.130) already suggests that the ˆa(k) and ˆa^†(k) operators are precisely the single-quantum destruction and creation operators for the continuum problem. To verify this interpretation and find the eigenvalues of Hˆ , we now insert the expansion for φˆ and ˆπ into H ˆ of (5.124). One finds the remarkable result (problem 5.7)

∫ _∞ ( )

Hˆ = dk 1

[ˆa ^†(k)ˆa(k) + ˆa(k)ˆa ^†(k)]ω . (5.132) 2π 2

−∞

Comparing this with the single-oscillator result

Hˆ = (ˆ1₂ a ^†aˆ + ˆaaˆ^†)ω (5.133) shows that, as anticipated in section 5.1, each classical mode of the field can be quantized, and behaves like a separate oscillator coordinate, with its own frequency ω = k. The operator ˆa^†(k) creates, and ˆa(k) destroys, a quantum of the k mode. The factor (2π)⁻¹ in Hˆ arises from our normalization choice.

We note that in the field operator φˆ of (5.129), those terms which destroy quanta go with the factor e^−iωt, while those which create quanta go with

+iωt

e . This choice is deliberate and is consistent with the ‘absorption’ and

‘emission’ factors e^±iωt of ordinary time-dependent perturbation theory in quantum mechanics (cf equation (A.33) of appendix A).

What is the mass of these quanta? We know that their frequency ω is related to their wavenumber k by (5.126), which – restoring ħ’s and c’s – can be regarded as equivalent to ħω = ħck, or E = cp, where we use the Einstein

141 5.2. The quantumfield: (ii) Lagrange–Hamilton formulation

and de Broglie relations. This is precisely the E–p relation appropriate to a massless particle, as expected.

What is the energy spectrum? We expect the ground state to be deter­

mined by the continuum analogue of a

ˆ_r|0> = 0 for all r; (5.134) namely

a

ˆ(k)|0> = 0 for all k. (5.135) However, there is a problem with this. If we allow the Hamiltonian of (5.132) to act on |0> the result is not (as we would expect) zero, because of the a

ˆ(k)ˆa^†(k) term (the other term does give zero by (5.135)). In the single oscillator case, we rewrote ˆaaˆ in terms of ˆ† a^†aˆ by using the commutation relation (5.72), and this led to the ‘zero-point energy’, ¹₂ω, of the oscillator ground state. Adopting the same strategy here, we write Hˆ of (5.132) as

∫ ∫

dk dk 1

Hˆ = aˆ^†(k)ˆa(k)ω + [ˆa(k), aˆ^†(k)]ω. (5.136)

2π 2π 2

Now consider Hˆ|0>: we see from the definition of the vacuum (5.135) that the first term will give zero as expected – but the second term is infinite, since the commutation relation (5.130) produces the infinite quantity ‘δ(0)’ as k → k ^′ ; moreover, the k integral diverges.

This term is obviously the continuum analogue of the zero-point energy¹₂ω – but because there are infinitely many oscillators, it is infinite. The conven­

tional ploy is to argue that only energy differences, relative to a conveniently defined ground state, really matter – so that we may discard the infinite con­

stant in (5.136). Then the ground state |0> has energy zero, by definition, and the eigenvalues of Hˆ are of the form

∫ dk

n(k)ω (5.137)

2π

wheren(k) is the number of quanta (counted by the number operatoraˆ^†(k)ˆa(k)) of energy ω = k. For each definite k, and hence ω, the spectrum is like that of the simple harmonic oscillator. The process of going from (5.132) to (5.136) without the second term is called ‘normally ordering’ the ˆa and ˆa^† operators:

in a ‘normally ordered’ expression, all ˆa^†’s are to the left of all ˆa’s, with the result that the vacuum value of such expressions is by definition zero.

It has to be admitted that the argument that only energy differences matter is false as far as gravity is concerned, which couples to all sources of energy.

It would ultimately be desirable to have theories in which the vacuum energy came out finite from the start (as actually happens in ‘supersymmetric’ field theories – see for example Weinberg (1995), p 325); see also comment (3).

We proceed on to the excited states. Any desired state in which excitation quanta are present can be formed by the appropriate application of ˆa^†(k) op­

erators to the ground state |0>. For example, a two-quantum state containing

one quantum of momentum k1and another of momentum k2 may be written (cf (5.81))

|k1, k2> ∝aˆ^†(k1)ˆa ^†(k2)|0>. (5.138) A general state will contain an arbitrary number of quanta.

Once again, and this time more formally, we have completed the pro­

gramme outlined in section 5.1, ending up with the ‘quantization’ of a classical field φ(x, t), as exemplified in the basic expression (5.129), together with the interpretation of the operators aˆ(k) and aˆ^†(k) as destruction and creation op­

erators for mode quanta. We have, at least implicitly, still retained up to this point the ‘mechanical model’ of some material object oscillating – some kind of infinitely extended ‘jelly’. We now throw away the mechanical props and embrace the unadorned quantum field theory! We do not ask what is waving, we simply postulate a field – such as φ – and quantize it. Itsquantaofexcita­

tionarewhat wecall particles – for example, photons in the electromagnetic case.

We end this long section with some further remarks about the formalism, and the physical interpretation of our quantum field φˆ.

Comment (1)

The alert reader, who has studied appendix I, may be worried about the following (possible) consistency problem. The fields φˆ and ˆπ are Heisenberg picture operators, and obey the equations of motion

φˆ˙

(x, t) = −i[φˆ(x, t), Hˆ ] (5.139)

˙ ˆ

πˆ(x, t) = −i[ˆπ(x, t), H] (5.140) where Hˆ is given by (5.132). It is a good exercise to check (problem 5.8(a)) that (5.139) yields just the expected relation φ(x, t) = ˆ˙ˆ π(x, t) (cf (5.122)).

Thus (5.140) becomes

¨ ˆ ˆ

φ(x, t) = −i[ˆπ(x, t), H]. (5.141) However, we have assumed in our work here that φˆ obeyed the wave equation (cf.(5.121))

∂²

¨ ˆ ˆ

φ = φ(x, t) (5.142)

∂x²

as a consequence of the quantized version of the Euler–Lagrange equation (5.96).

Thus the right-hand sides of (5.141) and (5.142) need to be the same, for con­

sistency – and they are: see problem 5.8(b). Thus – at least in this case – the Heisenberg operator equations of motion are consistent with the Euler–

Lagrange equations.

Comment (2)

Following on from this, we may note that this formalism encompasses both the wave and the particle aspects of matter and radiation. The former is evi­

143 5.2. The quantumfield: (ii) Lagrange–Hamilton formulation

dent from the plane-wave expansion functions in the expansion of φˆ, (5.129), which in turn originate from the fact that φˆ obeys the wave equation (5.121).

The latter follows from the discrete nature of the energy spectrum and the associated operators ˆa, ˆa^† which refer to individual quanta i.e. particles.

Comment (3)

Next, we may ask: what is the meaning of the ground state |0>for a quantum field? It is undoubtedly the state with n(k) = 0 for all k, i.e. the state with no quanta in it – and hence no particles in it, on our new interpretation. It is therefore the vacuum! As we shall see later, this understanding of the vacuum as the ground state of a field system is fundamental to much of modern particle physics – for example, to quark confinement and to the generation of mass for the weak vector bosons. Note that although we discarded the overall (infinite) constant in Hˆ , differences in zero-point energies can be detected; for example, in the Casimir effect (Casimir 1948, Kitchener and Prosser 1957, Sparnaay 1958, Lamoreaux 1997, 1998). These and other aspects of the quantum field theory vacuum are discussed in Aitchison (1985).

Comment (4)

Consider the two-particle state (5.138): |k1, k2> ∝aˆ^†(k1)ˆa^†(k2)|0>. Since the a

ˆ^† operators commute, (5.130), this state is symmetric under the interchange k1 ↔k2. This is an inevitable feature of the formalism as so far developed – there is no possible way of distinguishing one quantum of energy from another, and we expect the two-quantum state to be indifferent to the order in which the quanta are put in it. However, this has an important implication for the particle interpretation: since the state is symmetric under interchange of the particle labels k₁ and k₂, it must describe identical bosons. How the formalism is modified in order to describe the antisymmetric states required for two fermionic quanta will be discussed in section 7.2.

Comment (5)

Finally, the reader may well wonder how to connect the quantum field theory formalism to ordinary ‘wavefunction’ quantum mechanics. The ability to see this connection will be important in subsequent chapters and it is indeed quite simple. Suppose we form a state containing one quantum of the φˆ field, with momentum k ^′ :

|k ^′ >= N aˆ^†(k ^′ )|0> (5.143) whereN is a normalization constant. Now consider the amplitude <0|φˆ(x, t)|k ^′ >. We expand this out as

∫ dk

<0|φˆ(x, t)|k ^′ >= <0| √ [ˆa(k)e^ikx−iωt + ˆa ^†(k)e^−ikx+iωt]N aˆ^†(k ^′ )|0>. 2π 2ω

(5.144)

The ‘ˆa^†aˆ^†’ term will give zero since <0|aˆ^† = 0. For the other term we use the commutation relation (5.130) to write it as

∫ Ndk eik ^' x−iω ^' t

<0| √ [ˆa ^†(k ^′ )ˆa(k) + 2πδ(k−k ^′ )]e^ikx−iωt|0> = N √ (5.145)

2π 2ω 2ω ^′

using the vacuum condition once again, and integrating over the δ function using the property (5.116) which sets k = k ^′ and hence ω = ω ^′ . The vacuum is normalized to unity, <0|0> = 1. The normalization constant N can be adjusted according to the desired convention for the normalization of the states and wavefunctions. The result is just the plane-wave wavefunction for a particle in the state |k ^′ >! Thus we discover that the vacuum to one-particle matrix elements of the field operators are just the familiar wavefunctions of single-particle quantum mechanics. In this connection we can explain some common terminology. The path to quantum field theory that we have followed is sometimes called ‘second quantization’ – ordinary single-particle quantum mechanics being the first-quantized version of the theory.

Im Dokument GAUGE THEORIES (Seite 153-160)