CURVED SPACETIME

=

µ 1

2πiΛ (τ⁰−τ)

¶D/2

exp

"

i 2Λ

(x⁰−x)² τ⁰−τ

#

, (1.75)

whereDis the dimension of spacetime (D= 4 in our case). In performing the above functional integration there is no need for ghosts. The latter are necessary in a constrained theory in order to cancel out the unphysical states related to reparametrizations. There are no unphysical states of such a kind in the unconstrained theory.

In eq. (1.75)K(τ, x;τ⁰, x⁰) is the probability amplitude to find the particle in a spacetime pointx^µat a value of the evolution parameterτ when it was previously found in the pointx^0µ atτ⁰. The propagator has also the role of the Green’s function satisfying the equation

µ i ∂

∂τ −H

¶

K(τ, x;τ⁰, x⁰) =δ(τ−τ⁰)δ^D(x−x⁰). (1.76) Instead ofK(τ, x;τ⁰, x⁰) we can use its Fourier transform

∆(µ, x−x⁰) = Z

dτ e^iµ2ΛτK(τ, x;τ⁰, x⁰), (1.77) which is just (1.74) withc(p) = 1,x^µandτ being replaced byx^µ−x^0µ and τ−τ⁰, respectively.

the volume ^p|g|dx⁰ in the neighborhood of the point x⁰ at the evolution time τ. The probability to find the particle anywhere in spacetime is 1,

hence⁵ _{Z q}

|g|dx ψ^∗(τ, x)ψ(τ, x) = 1. (1.80) Multiplying (1.79) byhx⁰⁰|(that is projecting the state|ψiinto an eigenstate hx⁰⁰|) we have

hx⁰⁰|ψi=ψ(τ, x⁰⁰) = Z

hx⁰⁰|x⁰i^q|g(x⁰)|dxhx⁰|ψi. (1.81) This requires the following normalization condition for the eigenvectors

hx⁰|x⁰⁰i= δ(x⁰−x⁰⁰)

p|g(x⁰)| = δ(x⁰−x⁰⁰)

p|g(x⁰⁰)| ≡δ(x⁰, x⁰⁰). (1.82) From (1.82) we derive

∂_µ⁰δ(x⁰, x⁰⁰) =−∂_µ⁰⁰δ(x⁰, x⁰⁰)− 1

p|g(x⁰⁰)|∂_µ^00q|g(x⁰⁰)|δ(x⁰, x⁰⁰), (1.83) (x^0µ−x^00µ)∂⁰_αδ(x⁰, x⁰⁰) =−δ^µ_αδ(x⁰, x⁰⁰). (1.84) We can now calculate the matrix elementshx⁰|p_µ|x⁰⁰i according to

hx⁰|[x^µ, p_ν]|x⁰⁰i= (x^0µ−x^00µ)hx⁰|p_ν|x⁰⁰i=iδ^µ_νδ(x⁰, x⁰⁰). (1.85) Using the identities (1.83), (1.84) we find

hx⁰|p_µ|x⁰⁰i= (−i∂_µ⁰ +F_µ(x⁰))δ(x⁰, x⁰⁰) (1.86) whereF_µ(x⁰) is an arbitrary function. If we take into account the commu-tation relations [p_µ, p_ν] = 0 and the Hermitian condition

hx⁰|p_µ|x⁰⁰i=hx⁰⁰|p_µ|x⁰i^∗ (1.87) we find thatF_µ is not entirely arbitrary, but can be of the form

F_µ(x) =−i|g|^−1/4∂_µ|g|^1/4. (1.88) Therefore

p_µψ(x⁰)≡ hx⁰|p_µ|ψi = Z

hx⁰|p_µ|x⁰⁰i^q|g(x⁰⁰)|dx⁰⁰hx⁰⁰|ψi

= −i(∂µ+|g|^−1/4∂µ|g|^1/4)ψ(x⁰) (1.89)

5We shall often omit the prime when it is clear that a symbol without a prime denotes the eigenvalues of the corresponding operator.

or

p_µ=−i(∂_µ+|g|^−1/4∂_µ|g|^1/4) (1.90) p_µψ=−i|g|^−1/4∂_µ(|g|^1/4ψ). (1.91) The expectation value of the momentum operator is

hp_µi= Z

ψ^∗(x⁰)^q|g(x⁰)|dx⁰hx⁰|p_µ|x⁰⁰i^q|g(x⁰⁰)|dx⁰⁰ψ(x⁰⁰)

=−i^{Z q}|g|dx ψ^∗(∂_µ+|g|^−1/4∂_µ|g|^1/4)ψ. (1.92) Because of the Hermitian condition (1.87) the expectation value of p_µ is real. This can be also directly verified from (1.92):

hp_µi^∗ = i^{Z q}|g|dx ψ(∂_µ+|g|^−1/4∂_µ|g|^1/4)ψ^∗

= −i Z q

|g|dx ψ^∗(∂µ+|g|^−1/4∂µ|g|^1/4)ψ +i

Z

dx ∂_µ( q

|g|ψ^∗ψ). (1.93)

The surface term in (1.93) can be omitted and we findhp_µi^∗ =hp_µi. Point transformations In classical mechanics we can transform the generalized coordinates according to

x^0µ=x^0µ(x). (1.94)

The conjugate momenta then transform as covariant components of a vec-tor:

p⁰_µ= ∂x^ν

∂x^0µp_ν. (1.95)

The transformations (1.94), (1.95) preserve the canonical nature ofx^µ and p_µ, and define what is called a point transformation.

According to DeWitt, point transformations may also be defined in quantum mechanics in an unambiguous manner. The quantum analog of eq. (1.94) retains the same form. But in eq. (1.95) the right hand side has to be symmetrized so as to make it Hermitian:

p⁰_µ= ¹₂ µ∂x^ν

∂x^0µp_ν+p_ν ∂x^ν

∂x^0µ

¶

. (1.96)

Using the definition (1.90) we obtain by explicit calculation

p⁰_µ=−i^³∂_µ⁰ +|g⁰|^−1/4∂_µ⁰|g⁰|^1/4´. (1.97)

Expression (1.90) is therefore covariant under point transformations.

Quantum dynamical theory is based on the following postulate:

The temporal behavior of the operators representing the observables of a physical system is determined by the unfolding-in-time of a unitary trans-formation.

‘Time’ in the above postulate stands for the evolution time (the evolution parameterτ). As in flat spacetime, the unitary transformation is

U =e^iHτ (1.98)

where the generator of evolution is the Hamiltonian H= Λ

2(p^µp_µ−κ²). (1.99) We shall consider the case when Λ and the metricg_µν are independent of τ.

Instead of using the Heisenberg picture in which operators evolve, we can use the Schr¨odinger picture in which the states evolve:

|ψ(τ)i=e^−iHτ|ψ(0)i. (1.100) From (1.98) and (1.100) we have

i∂|ψi

∂τ =H|ψi, i∂ψ

∂τ =Hψ, (1.101)

which isthe Schr¨odinger equation. It governs the state or the wave function defined over the entire spacetime.

In the expression (1.99) for the Hamiltonian there is an ordering ambigu-ity in the definition ofp^µpµ =gµνp^µp^ν. Since pµ is a differential operator, it matters at which place one putsg_µν(x); the expressiong_µνp^µp^ν is not the same asp^µg_µνp^ν orp^µp^νg_µν.

Let us use the identity

|g|^1/4p_µ|g|^−1/4 =−i ∂_µ (1.102) which follows immediately from the definition (1.90), and define

p^µp_νψ = |g|^−1/2³|g|^1/4p_µ|g|^−1/4´|g|^1/2g^µν³|g|^1/4p_ν|g|^−1/4´ψ

= −|g|^−1/2∂_µ(|g|^1/2g^µν∂_νψ)

= −DµD^µψ, (1.103)

where D_µis covariant derivative with respect to the metricg_µν. Expression (1.103) is nothing but an ordering prescription: if one could neglect that p_µand g_µν (or|g|) do not commute, then the factors|g|^−1/2,|g|^1/4, etc., in eq. (1.103) altogether would give 1.

One possible definition of the square of the momentum operator is thus p²ψ=|g|^−1/4p_µg^µν|g|^1/2p_ν|g|^−1/4ψ=−D_µD^µψ. (1.104) Another well known definition is

p²ψ= ¹₄(pµpνg^µν+ 2pµg^µνpν+g^µνpµpν)ψ

= (−DµD^µ+¹₄R+g^µνΓ^α_βµΓ^β_αν)ψ. (1.105) Definitions (1.104), (1.105) can be combined according to

p²ψ= ¹₆(2|g|^−1/4p_µg^µν|g|^1/2p_ν|g|^−1/4+p_µp_νg^µν+ 2p_µg^µνp_ν +g^µνp_µp_ν)ψ

= (−D_µD^µ+¹₆R+²₃g^µνΓ^α_βµΓ^β_αν)ψ. (1.106) One can verify that the above definitions all give the Hermitian operatorsp². Other, presumably infinitely many, Hermitian combinations are possible.

Because of such an ordering ambiguity the quantum Hamiltonian H = Λ

2p² (1.107)

is undetermined up to the terms like (¯h²κ/2)(R+4g^µνΓ^α_βµΓ^β_αν), (withκ= 0,

1

4, ¹₆, etc., which disappear in the classical approximation where ¯h→0).

The Schr¨odinger equation (1.101) admits the following continuity

equa-tion ∂ρ

∂τ + D_µj^µ= 0 (1.108)

whereρ=ψ^∗ψand j^µ= Λ

2 [ψ^∗p^µψ+ (ψ^∗p^µψ)^∗] =−iΛ

2(ψ^∗∂^µψ−ψ ∂^µψ^∗). (1.109)

The stationary Schr¨odinger equation. If we take the ansatz ψ=e^−iEτφ(x), (1.110) whereE is a constant, then we obtain the stationary Schr¨odinger equation

Λ

2(−D_µD^µ−κ²)φ=Eφ, (1.111)

which has the form of the Klein–Gordon equation in curved spacetime with squared massM² =κ²+ 2E/Λ.

Let us derive the classical limit of eqs. (1.91) and (1.101). For this purpose we rewrite those equations by including the Planck constant ¯h⁶:

ˆ

pµψ=−i¯h|g|^−1/4∂µ(|g|^1/4ψ), (1.112) Hψˆ =i¯h∂ψ

∂τ. (1.113)

For the wave function we take the expression ψ=A(τ, x)exp

·i

¯

hS(τ, x)

¸

(1.114) with realAand S.

Assuming (1.114) and taking the limit ¯h→0, eq. (1.112) becomes ˆ

pµψ=∂µS ψ. (1.115)

Inserting (1.114) into eq. (1.113), taking the limit ¯h → 0 and writing separately the real and imaginary part of the equation, we obtain

−∂S

∂τ = Λ

2 (∂_µS∂^µS−κ²), (1.116)

∂A²

∂τ + D_µ(A²∂^µS) = 0. (1.117) Equation (1.116) is just the Hamilton–Jacobi equation of the classical point particle theory in curved spacetime, whereE =−∂S/∂τ is ”energy”

(the spacetime analog of the non-relativistic energy) and p_µ = ∂_µS the classical momentum.

Equation (1.117) is the continuity equation, where ψ^∗ψ = A² is the probability densityand

A²∂^µS =j^µ (1.118)

isthe probability current.

6Occasionally we use the hat sign over the symbol in order to distinguish operators from their eigenvalues.

The expectation value. If we calculate the expectation value of the momentum operatorpµwe face a problem, since the quantityhpµiobtained from (1.92) is not, in general, a vector in spacetime. Namely, if we insert the expression (1.114) into eq. (1.92) we obtain

Z q

|g|dx ψ^∗p_µψ = ^{Z q}|g|dx A²∂_µS−i¯h^{Z q}|g|dx(A∂_µA +A²|g|^−1/4∂_µ|g|^1/4

= ^{Z q}|g|dx A²∂_µS−i¯h 2

Z

dx ∂_µ(^q|g|A²)

= Z q

|g|dx A²∂_µS, (1.119) where in the last step we have omitted the surface term.

If, in particular, we choose a wave packet localized around a classical trajectoryx^µ=X_c^µ(τ), then for a certain period ofτ (until the wave packet spreads too much), the amplitude is approximately

A²= δ(x−X_c)

p|g| . (1.120)

Inserting (1.120) into (1.119) we have

hpµi=∂µS|_Xc =pµ(τ) (1.121) That is, the expectation value is equal to the classical momentum 4-vector of the center of the wave packet.

But in general there is a problem. In the expression

hp_µi=^{Z q}|g|dx A²∂_µS (1.122) we integrate a vector field over spacetime. Since one cannot covariantly sum vectors at different points of spacetime,hp_µiis not a geometric object:

it does not transform as a 4-vector. The result of integration depends on which coordinate system (parametrization) one chooses. It is true that the expression (1.119) is covariant under the point transformations (1.94), (1.96), but the expectation value is not a classical geometric object: it is neither a vector nor a scalar.

One way to resolve the problem is in consideringhp_µias a generalized geo-metric object which is a functional of parametrization. If the parametriza-tion changes, then also hpµi changes in a well defined manner. For the

ansatz (1.114) we have

hp⁰_µi=^{Z q}|g⁰|dx⁰A⁰²(x⁰)∂⁰_µS=^{Z q}|g|dx A²(x)∂x^ν

∂x^0µ∂_νS. (1.123) The expectation value is thus a parametrization dependent generalized ge-ometric object. The usual classical gege-ometric objectp_µis also parametriza-tion dependent, since it transforms according to (1.95).

Product of operators.. Let us calculate the product of two operators.

We have

hx|pµpν|ψi = Z

hx|pµ|x⁰i^q|g(x⁰)|dx⁰hx⁰|pν|ψi

= Z

(−i)(∂_µ+¹₂Γ^α_µα)δ(x−x⁰)

p|g(x⁰)| (−i)∂_ν⁰ψ^q|g(x⁰)|dx⁰

= −∂_µ∂_νψ−¹₂Γ^α_µα∂_νψ, (1.124) where we have used the relation

1

|g|^1/4 ∂_µ|g|^1/4 = 1

2^p|g|∂_µ^q|g|= ¹₂Γ^α_µα. (1.125) Expression (1.124) is covariant with respect to point transformations, but it is not a geometric object in the usual sense. If we contract (1.124) by g^µν we obtain g^µν(−∂µ∂νψ − ¹₂Γ^α_µα∂νψ) which is not a scalar under reparametrizations (1.94). Moreover, as already mentioned, there is an ordering ambiguity where to insertg^µν.

FORMULATION IN TERMS OF A LOCAL LORENTZ FRAME

Components p_µ are projections of the vector p into basis vectors γ_µ of a coordinate frame. A vector can be expanded as p = pνγ^ν and the components are given by p_µ = p ·γ_µ (where the dot means the scalar product⁷). Instead of acoordinate framein which

γ_µ·γ_ν =g_µν (1.126)

we can use a local Lorentz frameγa(x) in which

γ_a·γ_b=η_ab (1.127)

whereη_ab is the Minkowski tensor. The scalar product

γ^µ·γ_a=e_a^µ (1.128)

7For a more complete treatment see the section on Clifford algebra.

isthe tetrad, orvierbein, field.

With the aid of the vierbein we may define the operators

pa= ¹₂(eaµpµ+pµeaµ) (1.129) with matrix elements

hx|p_a|x⁰i=−i Ã

e_a^µ∂_µ+ 1

2^p|g|∂_µ(^q|g|e_a^µ)

!

δ(x, x⁰) (1.130) and

hx|p_a|ψi=−i Ã

e_a^µ∂_µ+ 1

2^p|g|∂_µ(^q|g|e_a^µ)

!

.ψ (1.131)

In the coordinate representation we thus have

pa=−i(eaµ∂µ+ Γa), (1.132) where

Γ_a≡ 1

2^p|g|∂_µ(^q|g|e_a^µ). (1.133) One can easily verify that the operator p_a is invariant under the point transformations (1.95).

The expectation value

hp_ai= Z

dx^q|g|ψ^∗p_aψ (1.134) has some desirable properties. First of all, it is real, hp_ai^∗ = hp_ai, which reflects that p_a is a Hermitian operator. Secondly, it is invariant under coordinate transformations (i.e., it transforms as a scalar).

On the other hand, since p_a depends on a chosen local Lorentz frame, the integral hpai is a functional of the frame field eaµ(x). In other words, the expectation value is an object which is defined with respect to a chosen local Lorentz frame field.

Inflat spacetime we can choose a constant frame fieldγ_a, and the com-ponentse_a^µ(x) are then the transformation coefficients into a curved coor-dinate system. Eq. (1.134) then means that after having started from the momentump_µ in arbitrary coordinates, we have calculated its expectation value in a Lorentz frame. Instead of a constant frame fieldγa, we can choose anx-dependent frame field γ_a(x); the expectation value of momentum will then be different from the one calculated with respect to the constant frame field. In curved spacetime there is no constant frame field. First one has to choose a frame field γ_a(x), then define componentsp_a(x) in this frame field, and finally calculate the expectation value hpai.

Let us now again consider the ansatz (1.114) for the wave function. For the latter ansatz we have

hp_ai= Z

dx^q|g|A²e_a^µ∂_µS, (1.135) where the term with total divergence has been omitted. If the wave packet is localized around a classical world lineX_c(τ), so that for a certainτ-period A² =δ(x−Xc)/^p|g|is a sufficiently good approximation, eq. (1.135) gives hpˆ_ai=e_a^µ∂_µS|_Xc(τ)=p_a(τ), (1.136) wherep_a(τ) is the classical momentum along the world line X_c^µ in a local Lorentz frame.

Now we may contract (1.136) by e^a_µ and we obtain p_µ(τ) ≡ e^a_µp_a(τ), which are components of momentum alongX_c^µ in a coordinate frame.

Product of operators. The product of operators in a local Lorentz frame is given by

hx|p_ap_b|ψi = Z

hx|p_a|x⁰i^q|g(x⁰)|dx⁰hx⁰|p_b|ψi

= Z

(−i) (eaµ(x)∂µ+ Γa(x)) δ(x−x⁰) p|g(x⁰)|

q

|g(x⁰)|dx⁰

×(−i)^¡e_b^ν(x⁰)∂_ν⁰ + Γ_a(x⁰)^¢ψ(x⁰)

= −(e_a^µ∂_µ+ Γ_a)(e_b^ν∂_ν+ Γ_b)ψ=p_ap_bψ. (1.137) This can then be mapped into the coordinate frame:

e^a_αe^b_βp_ap_bψ=−e^q_αe^b_β(e^µ_a∂_µ+ Γ_a)(e^ν_b∂_ν+ Γ_b)ψ. (1.138) At this point let us use

∂_µe^a_ν−Γ^λ_µνe^a_λ+ω^a_bµe^b_ν = 0, (1.139) from which we have

ω^ab_µ=e^aνe^b_ν;µ (1.140) and

Γ_a≡ ¹₂ 1

p|g|∂_µ(^q|g|e_a^µ) = ¹₂e_a^µ_;µ= ¹₂ω_ca^µe^c_µ. (1.141) Here

e^a_ν;µ≡∂_µe^a_ν−Γ^λ_µνe^a_λ (1.142)

is the covariant derivative of the vierbein with respect to the metric.

The relation (1.139) is a consequence of the relation which tells us how the frame fieldγ^a(x) changes with position:

∂µγa=ω^a_bµγ^b. (1.143) It is illustrative to calculate the product (1.138) first in flat spacetime.

Then one can always choose a constant frame field, so thatω_abµ= 0. Then eq. (1.138) becomes

e^aαe^bβpapbψ=−e^bβ∂α(ebν∂νψ) =−∂α∂βψ+Γ^ν_αβ∂νψ=−DαDβψ, (1.144) where we have used

e^b_βe_b^ν =δ_b^ν , e^b_β,αe_b^ν =−e^b_βe_b^ν_,α, (1.145) and the expression for the affinity

Γ^µ_αβ =e^aµe_aα,β (1.146) which holds in flat spacetime where

e^a_µ,ν −e^a_ν,µ = 0. (1.147) We see that the product of operators is just the product of the covariant derivatives.

Let us now return to curved spacetime and consider the expression (1.138).

After expansion we obtain

−e^a_αe^b_βp_ap_bψ= D_αD_βψ+ω^ab_αe_aβe_b^ν∂_νψ+¹₂e^c_µω_ca^µ(e^a_α∂_βψ+e^a_β∂_αψ) +¹₂e^a_βe^c_µω_ca^µ_;αψ−¹₂e^a_βe^b_µω_ca^µω^c_bαψ

+¹₄e^a_αe^b_βe^c_µe^d_νω_ca^µω_db^νψ. (1.148) This is not a Hermitian operator. In order to obtain a Hermitian operator one has to take a suitable symmetrized combination.

The expression (1.148) simplifies significantly if we contract it by g^αβ and use the relation (see Sec. 6.1) for more details)

R=e_a^µe_b^ν(ω^ab_ν;µ−ω^ab_µ;ν+ω^ac_µω_c^b_ν −ω^ac_νω_c^b_µ). (1.149) We obtain

−g^αβe^aαe^b_βpap_bψ=−η^abpap_bψ=−p^apaψ

=^³D_αD^α−¹₄R−¹₄e^aµe_bνω_ca^νω^cb_µ^´ (1.150) which is a Hermitian operator. If one tries another possible Hermitian combination,

1 4

³e^aµpae^bµp_b+e^bµpae^aµp_b+pae^aµp_be^bµ+pae^bµp_be^aµ

´, (1.151) one finds, using [p_a, e^bµ] =−ie_a^ν∂_νe^bµ, that it is equal toη^abp_ap_b since the extra terms cancel out.

1.4. SECOND QUANTIZATION

In the first quantized theory we arrived at the Lorentz invariant Schr¨odin-ger equation. Now, following refs. [8]–[17], [19, 20] we shall treat the wave functionψ(τ, x^µ) as a field and the Schr¨odinger equation as a field equation which can be derived from an action. First, we shall consider the classical field theory, and then the quantized field theory of ψ(τ, x^µ). We shall construct a Hamiltonian and find out that the equation of motion is just the Heisenberg equation. Commutation relations for our field ψ and its conjugate canonical momentum iψ^† are defined in a straightforward way and are not quite the same as in the conventional relativistic field theory.

The norm of our states is thus preserved, although the states are localized in spacetime. Finally, we point out that for the states with definite masses the expectation value of the energy–momentum and the charge operator coincide with the corresponding expectation values of the conventional field theory. We then compare the new theory with the conventional one.

CLASSICAL FIELD THEORY WITH INVARIANT

In document The Landscape of Theoretical Physics: A Global View (Strani 39-50)