THE CANONICAL QUANTIZATION

whereδ/δψ(x⁰⁰), δ/δΠ(x⁰⁰) are functional derivatives.

If in eq. (1.171) we put A = ψ and take into account eq. (1.155) with

¯δψ= 0, we obtain

∂ψ

∂τ =− {ψ, H}, (1.173)

∂µψ=− {ψ, Hµ}. (1.174) Eq.(1.173) is equivalent to the Schr¨odinger equation (1.67).

A generic fieldψ which satisfies the Schr¨odinger equation (1.67) can be written as a superposition of the states with definitep_µ:

ψ(τ, x) = Z

d^Dp c(p) exp [ipµx^µ] exp

·

−iΛ

2(p²−κ²)τ

¸

. (1.175)

Using (1.175) we have H =−Λ

2 Z

d^Dp(p²−κ²)c^∗(p)c(p), (1.176) H_µ=

Z

d^Dp p_µc^∗(p)c(p). (1.177) On the other hand, from the spacetime stress–energy tensor (1.160) we find the total energy–momentum

P^µ= Z

dΣ_νT^µν. (1.178)

It is not the generator of spacetime translations. Use of P^µ as generator of spacetime translations results in all conceptual and technical complica-tions (including the lack of manifest Lorentz covariance) of the conventional relativistic field theory.

or

[ψ(τ, x), ψ^†(τ⁰, x⁰)]|τ⁰=τ =δ(x−x⁰), (1.180) and

[ψ(τ, x), ψ(τ⁰, x⁰]|τ⁰=τ = [ψ^†(τ, x), ψ^†(τ⁰, x⁰)]|τ⁰=τ = 0. (1.181) In momentum space the above commutation relations read

[c(p), c^†(p⁰)] =δ(p−p⁰), (1.182) [c(p), c(p⁰)] = [c^†(p), c^†(p⁰)] = 0. (1.183) The general commutator (forτ⁰ 6=τ) is:

[ψ(τ, x), ψ^†(τ⁰, x⁰)] = Z

d^Dp e^{ipµ(xµ−x0µ)}e^{−Λ2(p2−κ2)(τ−τ0)}. (1.184) The operatorsψ^†(τ, x) and ψ(τ, x) are creation and annihilation opera-tors, respectively. The vacuum is defined as a state which satisfies

ψ(τ, x)|0i= 0. (1.185) If we act on the vacuum by ψ^†(τ, x) we obtain a single particle state with definite position:

ψ^†(τ, x)|0i=|x, τi. (1.186) An arbitrary 1-particle state is a superposition

|Ψ⁽¹⁾i= Z

dx f(τ, x)ψ^†(τ, x)|0i, (1.187) wheref(τ, x) is the wave function.

A 2-particle state with one particle atx1 and another atx2 is obtained by applying the creation operator twice:

ψ^†(τ, x₁)ψ^†(τ, x₂)|0i=|x₁, x₂, τi. (1.188) In general

ψ^†(τ, x₁)...ψ^†(τ, x_n)|0i=|x₁, ..., x_n, τi, (1.189) and an arbitraryn-particle state is the superposition

|Ψ⁽ⁿ⁾i= Z

dx₁...dx_nf(τ, x₁, ..., x_n)ψ^†(τ, x₁)...ψ^†(τ, x_n)|0i, (1.190)

wheref(τ, x₁, ..., x_n) is the wave function thenparticles are spread with.

In momentum space the corresponding equations are

c(p)|0i= 0, (1.191)

c^†(p)|0i=|pi, (1.192) c^†(p₁)...c^†(p_n)|0i=|p₁, ..., p_ni, (1.193)

|Ψ⁽ⁿ⁾i= Z

dp₁...dp_ng(τ, p₁, ..., p_n)c^†(p₁)...c^†(p_n)|0i (1.194) The most general state is a superposition of the states|Ψ⁽ⁿ⁾iwith definite numbers of particles.

According to the commutation relations (1.180)–(1.183) the states

|x₁, ..., x_niand |p₁, ..., p_ni satisfy

hx1, ..., xn|x⁰₁, ..., x⁰_ni=δ(x1−x⁰₁)...δ(xn−x⁰_n), (1.195) hp1, ..., pn|p⁰₁, ..., p⁰_ni=δ(p1−p⁰₁)...δ(pn−p⁰_n), (1.196) which assures that the norm of an arbitrary state|Ψiis always positive.

We see that the formulation of the unconstrained relativistic quantum field theory goes along the same lines as the well known second quantization of a non-relativistic particle. Instead of the 3-dimensional space we have now a D-dimensional space whose signature is arbitrary. In particular, we may take a 4-dimensional space with Minkowski signature and identify it withspacetime.

THE HAMILTONIAN AND THE GENERATOR OF SPACETIME TRANSLATIONS

The generatorG(τ) defined in (1.168) is now an operator. An infinites-imal change of an arbitrary operatorA is given by the commutator

δA=−i[A, G(τ)], (1.197) which is the quantum analog of eq. (1.171). If we take A = ψ, then eq. (1.197) becomes

∂ψ

∂τ =i[ψ, H], (1.198)

∂ψ

∂x^µ =i[ψ, Hµ], (1.199) where

H =

Z

d^DxΘ =−Λ 2 Z

d^Dx(∂_µψ^†∂^µψ−κ²ψ^†ψ), (1.200) Hµ =

Z

d^DxΘµ=−i Z

d^Dx ψ^†∂µψ. (1.201) Using the commutation relations (1.179)–(1.181) we find that eq. (1.198) is equivalent to the field equation (1.67) (the Schr¨odinger equation). Eq.

(1.198) is thus the Heisenberg equation for the field operator ψ. We also find that eq. (1.199) gives just the identity∂_µψ=∂_µψ.

In momentum representation the field operators are expressed in terms of the operatorsc(p), c^†(p) according to eq. (1.175) and we have

H = −Λ 2

Z

d^Dp(p²−κ²)c^†(p)c(p₎, (1.202) H_µ =

Z

d^Dp p_µc^†(p)c(p). (1.203) The operator H is the Hamiltonian and it generates the τ-evolution, whereas H_µ is the generator of spacetime translations. In particular, H₀ generates translations along the axisx⁰ and can be either positive or neg-ative definite.

ENERGY–MOMENTUM OPERATOR

Let us now consider the generatorG(Σ) defined in eq. (1.170) with T^µν

given in eq. (1.160) in which the classical fields ψ,ψ^∗ are now replaced by the operatorsψ, ψ^†. The total energy–momentum P_ν of the field is given by the integration ofT^µ_ν over a space-like hypersurface:

P_ν = Z

dΣ_µT^µ_ν. (1.204)

Instead ofP_ν defined in (1.204) it is convenient to introduce Pe_ν =

Z

ds P_ν, (1.205)

where ds is a distance element along the direction n^µ which is orthogonal to the hypersurface element dΣ_µ. The latter can be written as dΣ_µ = n_µdΣ. Using dsdΣ = d^Dx and integrating out x^µ in (1.205) we find that τ-dependence disappears and we obtain (see Box 1.1)

Pe_ν = Z

d^DpΛ

2(n_µp^µ)p_ν(c^†(p)c(p) +c(p)c^†(p)). (1.206)

Box 1.1

¾

?

¾

?

¾

?

¾

?

...

...

...

...

.........

............

...

...

......

......

G(Σ) =^RdΣ_µdτ T^µ_νδx^ν

δx^µ=ξ^µδs , ξ^µ a vector along δx^µ-direction dΣ_µ=n_µdΣ, n_µ a vector along dΣ_µ-direction

P(n, ξ) =^R dΣ dτ T^µ_νn_µξ^ν

integrate over dsand divide bys₂−s₁

P(n, ξ) = lim

s1,2→−∞,∞

1 s2−s1

Z _s2

s1

dΣ dsdτ T^µ_νn_µξ^ν

dΣ ds= d^Dx

P(n, ξ) = lim

s1,2→−∞,∞ τ1−τ2

s1−s2

Z _s2

s1

d^DpΛ

2 (n_µp^µ)p_νξ^ν³c^†(p)c(p) +c(p)c^†(p)^´ Λ = ^∆s_∆τ √¹

p² when mass is definite

P(n, ξ) = ¹₂^R d^Dp ²(np)p_νξ^ν³c^†(p)c(p) +c(p)c^†(p)^´

When ξ^ν is time-like P(n, ξ) is projection of energy, when ξ^ν is space-likeP(n, ξ) is projection ofmomentum on ξ^ν

From (1.206) we have that the projection ofP^e_ν on a time-like vector n^ν is always positive, while the projection on a space-like vector N^ν can be positive or negative, depending on signs of p_µn^µ and p_µN^µ. In a special reference frame in which n^ν = (1,0,0, ...,0) this means that P0 is always positive, whereasP_r,r= 1,2, ..., D−1, can be positive or negative definite.

What is the effect of the generator P^e_νδx^ν on a field ψ(τ, x). From (1.197), (1.206) we have

δψ = −i[ψ,P^eν]δx^ν (1.207)

= −i Z

d^DpΛ(n_µp^µ)c(p)exp [ip_µx^µ] exp

·

−iΛ

2 (p²−κ²)τ

¸ p_νδx^ν Acting on a Fourier component ofψ with definite pµp^µ=M² the gene-rator P^e_νδx^ν gives

δφ=−i[φ,P^e_ν]δx^ν =−i Z

d^DpΛM ²(np)c(p)e^ipµxµδ(p²−M²)p_νδx^ν

=−∂_νφ⁽⁺⁾δx^ν+∂_νφ⁽⁻⁾δx^ν, (1.208)

²(np)≡ n^µp_µ p|p²| =

½ +1, n^µp_µ>0

−1, n^µpµ<0 (1.209) where

φ(x) = Z _∞

−∞dτ e^iµτψ(τ, x)

= Z _∞

p0=−∞d^Dp δ(p²−M²)c(p)e^ipµxµ

= Z _∞

p0=0d^Dp δ(p²−M²)c(p)e^ipµxµ +

Z ₀

p0=−∞

d^Dp δ(p²−M²)c(p)e^ipµxµ

= φ⁽⁺⁾+φ⁽⁻⁾ (1.210)

As in Sec. 1.3. we identifyµ²+κ²≡M².

We see that the action of P^eνδx^ν on φ differs from the action of Hνδx^ν. The difference is in the step function ²(np). When acting on φ⁽⁺⁾ the generator P^e_νδx^ν gives the same result as H_νδx^ν. On the contrary, when acting on φ⁽⁻⁾ it gives the opposite sign. This demonstrates that P^eνδx^ν

does not generate translations and Lorentz transformations in Minkowski space MD. This is a consequence of the fact that P0 is always positive definite.

PHASE TRANSFORMATIONS AND THE CHARGE OPERATOR

Let us now consider the case when the field is varied at the boundary, whileδx^µand δτ are kept zero. Let the field transform according to

ψ⁰ =e^iαψ , δψ=iαψ , δψ^†=−iαψ^†. (1.211) Eq. (1.156) then reads

δI = I

dτdΣ_µ µ

−iΛα 2

¶

(∂^µψ^†ψ−∂^µψ ψ^†) + Z

d^Dx(−α)ψ^†ψ

¯¯

¯¯

¯

τ2

τ1

(1.212) Introducingthe charge density

ρc =−αψ^†ψ (1.213)

andthe charge current density j_c^µ=−iΛα

2 (∂^µψ^†ψ−∂^µψ ψ^†), (1.214) and assuming that the action is invariant under the phase transformations

(1.211) we obtain from (1.212) the following conservation law:

I

dΣµj_c^µ+ d dτ

Z

d^Dx ρc = 0, (1.215) or

∂_µj_c^µ+∂ρc

∂τ = 0. (1.216)

In general, when the field has indefinite mass it is localized in spacetime and the surface term in (1.215) vanishes. So we have that the generator

C =− Z

d^Dx ρ_c (1.217)

is conserved at allτ-values.

In particular, when ψ has definite mass, then ψ^†ψ is independent of τ and the conservation of C is trivial. Since ψ is not localized along a time-like direction the current density j_c^µ does not vanish at the space-like hypersurface even if taken at infinity. It does vanish, however, at a sufficiently far away time-like hypersurface. Therefore the generator

Q= Z

dΣ_µj_c^µ (1.218)

is conserved at all space-like hypersufaces. The quantityQcorresponds to the charge operator of the usual field theory. However, it is not generator of the phase transformations (1.211).

By using the field expansion (1.175) we find C =−α

Z

d^Dp c^†(p)c(p). (1.219) From

δψ=−i[ψ, C] (1.220)

and the commutation relations (1.179)–(1.183) we have

δψ=iαψ , δψ^†=−iαψ^†, (1.221) δc(p) =iαc(p), δc^†(p) =−iαc^†(p), (1.222) which are indeed the phase transformations (1.211). Therefore C is the generator of phase transformations.

On the other hand, from δψ = −i[ψ, Q] we do not obtain the phase transformations. But we obtain from Q something which is very close to the phase transformations, if we proceed as follows. Instead of Q we introduce

Qe = Z

ds Q, (1.223)

where dsis an infinitesimal interval along a time-like direction n^µ orthogo-nal to dΣ_µ=n_µdΣ. Using dsdΣ = d^Dx and integrating out x^µ eq. (1.223) becomes

Qe= Z µ

−αΛ 2

¶

dp(p^µn_µ)(c^†(p)c(p) +c(p)c^†(p)). (1.224) The generatorQ^e then gives

δψ=−i[ψ,Q] =^e Z

iαΛd^Dp(p^νnν)c(p)exp [ipµx^µ] exp

·

−iΛ

2 (p²−κ²)τ

¸ . (1.225) For a Fourier component with definitepµp^µ we have

δφ=−i[φ,Q] =^e Z

d^Dp iαΛm ²(np)c(p)e^ipµxµδ(p²−M²)

= iαφ⁽⁺⁾−iαφ⁽⁻⁾ (1.226)

where²(np) andφ⁽⁺⁾,φ⁽⁻⁾ are defined in (1.209) and (1.210), respectively.

Action of operators on states. From the commutation relations (1.180)–

(1.180) one finds that when the operators c^†(p) and c(p) act on the eigen-states of the operators H, Hµ, P^eµ, C, Q^e they increase and decrease, re-spectively, the corresponding eigenvalues. For instance, if H_µ|Ψi = p_µ|Ψi then

H_µc^†(p⁰)|Ψi= (p_µ+p⁰_µ)|Ψi, (1.227) H_µc(p⁰)|Ψi= (p_µ−p⁰_µ)|Ψi, (1.228) and similarly for other operators.

THE EXPECTATION VALUES OF THE OPERATORS

Our parametrized field theory is just a straightforward generalization of the non-relativistic field theory in E₃. Instead of the 3-dimensional Eu-clidean spaceE₃we have now the 4-dimensional Minkowski spaceM₄(or its D-dimensional generalization). Mathematically the theories are equivalent (apart from the dimensions and signatures of the corresponding spaces). A good exposition of the unconstrained field theory is given in ref. [35]

A state |Ψ_ni of n identical free particles can be given in terms of the generalized wave packet profiles g⁽ⁿ⁾(p₁, ..., p_n) and the generalized state vectors|p₁, ..., p_ni:

|Ψ_ni= Z

dp₁...dp_ng⁽ⁿ⁾(p₁, ..., p_n)|p₁, ..., p_ni. (1.229) The action of the operatorsc(p),c^†(p) on the state vector is the following:

c(p)|p₁, ..., p_ni = 1

√n Xn i=1

δ(p−p_i)|p₁, ..,ˇp_i, .., p_ni, (1.230) c^†(p)|p₁, ..., p_ni = √

n+ 1|p, p₁, ..., p_ni. (1.231) The symbolˇp_i means that the quantitypi is not present in the expression.

Obviously

|p₁, ..., p_ni= 1

√n!c^†(p_n)...c^†(p₁)|0i. (1.232) For the product of operators we have

c^†(p)c(p)|p₁, ..., p_ni= Xn i=1

δ(p−p_i)|p₁, ..., p_ni (1.233) The latter operator obviously counts how many there are particles with momentump. Integrating overp we obtain

Z

d^Dp c^†(p)c(p)|p₁, ..., p_ni=n|p₁, ..., p_ni. (1.234)

The expectation value of the operator P^eµ (see eq. (1.206)) in the state

|Ψni is

h|Ψ_n|P^eµ|Ψi = hΨ_n| Z

d^DpΛ

2(n_νp^ν)p^µ(2c^†(p)c(p) +δ(0))|Ψ_ni

= Λ Xn i=1

h(n_νp^ν_i)p^µ_ii+Λ 2δ(0)

Z

d^Dp(n_νp^ν)p^µ, (1.235) where

h(nνp^ν_i)p^µ_ii= Z

dp1...dpng^∗(n)(p1, ..., pn)g⁽ⁿ⁾(p1, ..., pn)(p^ν_inν)p^µ_i. In particular, the wave packet profile g⁽ⁿ⁾(p₁, ..., p_n) can be such that

|Ψ_ni=|p₁, ..., p_ni. Then the expectation value (1.235) becomes hp₁, ..., p_n|P^eµ|p₁, ..., p_ni= Λ

Xn i=1

(n_νp^ν_i)p^µ_i +Λ 2δ(0)

Z

d^Dp(n_νp^ν)p^µ. (1.236) The extra term ^Λ₂δ(0)^R d^Dp(nνp^ν)p^µ vanishes for a space-like component ofp^µ, but it does not vanish for a time-like component.

So far the momentap₁, ..., p_n of nidentical particles have not been con-strained to a mass shell. Let us now consider the case where all the momenta p₁, ..., p_n are constrained to the mass shell, so that p^µ_ip_iµ =p²_i =p² =M², i= 1,2, ..., n. From the classical equation of motionp^µ_i = ˙x^µ_i/Λ we have

Λ = dsi

dτ q1

p²_i

, (1.237)

where ds_i =^qdx^µ_idx_iµ. Now, Λ andτ being the same for all particles we see that ifp²_i is the same for all particles also the 4-dimensional speed dsi/dτ is the same. Inserting (1.237) into (1.236) and chosing parametrization such that ds_i/dτ = 1 we obtain the following expectation value of P^eµ in a state on the mass shell:

hP^eµim =hp₁, ..., p_n|P^eµ|p₁, ..., p_nim = Xn i=1

²(np_i)p^µ_i +δ(0) 2

Z

d^Dp ²(np)p^µ. (1.238) Takingn_ν = (1,0,0, ....,0) we have

hP^e0im= Xn i=1

²(p⁰_i)p⁰_i + δ(0) 2

Z

d^Dp ²(p⁰)p⁰, (1.239)

hP^erim= Xn i=1

²(p⁰_i)p^r_i , r = 1,2, ..., D−1. (1.240) So far we have considered momenta as continuous. However, if we imag-ine a large box and fix the boundary conditions, then the momenta are discrete. Then in eqs. (1.239), (1.240) particlesi= 1,2, ..., n do not neces-sarily all have different momentap^µ_i. There can ben_pparticles with a given discrete value of momentump. Eqs. (1.239), (1.240) can then be written in the form

hP^e0im = ^X

p

(n⁺_p +n⁻_p +1

2)ω_p, (1.241)

hP˜im = ^X

p

(n⁺_p +n⁻_p)p (1.242) Here

ω_p ≡ |^qp²+M²|=²(p⁰)p⁰,

whereas n⁺_p =n_ωp,p is the number of particles with positive p⁰ at a given value of momentum andn⁻_p =n_−ωp,−p . In eq. (1.242) we have used

− X∞

p=−∞

n_−ωp,pp= X∞

p=−∞

n_−ωp,−pp.

Instead of δ(0) in eq. (82) we have written 1, since in the case of discrete momentaδ(p−p⁰) is replaced byδ_pp⁰, and δ_pp= 1.

Equations (1.241), (1.242) are just the same expressions as obtained in the conventional, on shell quantized field theory of a non-Hermitian scalar field. In an analogous way we also find that the expectation value of the electric charge operator Q, when taken on the mass shell, is identical to^e that of the conventional field theory.

COMPARISON WITH THE CONVENTIONAL

In document The Landscape of Theoretical Physics: A Global View (Strani 54-64)