hPerim= Xn i=1
²(p0i)pri , 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 momentapµi. There can benpparticles with a given discrete value of momentump. Eqs. (1.239), (1.240) can then be written in the form
hPe0im = X
p
(n+p +n−p +1
2)ωp, (1.241)
hP˜im = X
p
(n+p +n−p)p (1.242) Here
ωp ≡ |qp2+M2|=²(p0)p0,
whereas n+p =nωp,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 byδpp0, and δpp= 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 toe that of the conventional field theory.
COMPARISON WITH THE CONVENTIONAL
which differs from zero both for time-like and for space-like separations between xµ and x0µ. In this respect the new theory differs significantly from the conventional theory in which the commutator is zero for space-like separations and which assures that the process
h0|φ(x)φ†(x0)|0i
has vanishing amplitude. No faster–than–light propagation is possible in the conventional relativistic field theory.
The commutation relations leading to the conventional field theory are [c(p), c†(p0)] =²(p0)δD(p−p0), (1.244) where²(p0) = +1 for p0 >0 and²(p0) =−1.
By a direct calculation we then find [φ(x), φ†(x0)] =
Z
dDp eip(x−x0)²(p0)δ(p2−M2)≡D(x−x0), (1.245) which is indeed different from zero when (x −x0)2 > 0 and zero when (x−x0)2 <0.
The commutation relations (1.244), (1.245) also assure that the Heisen-berg equation
∂φ
∂x0 =i[ψ, P0] (1.246) is equivalent to the field equation
(∂µ∂µ+M2)φ= 0. (1.247) Here P0, which now serves as Hamiltonian, is the 0-th component of the momentum operator defined in terms of the definite mass fieldsφ(x),φ†(x) (see Box 1.2).
If vacuum is defined according to
c(p)|0i= 0, (1.248)
then because of the commutation relation (1.244) the scalar product is hp|p0i=h0|c(p)c†|0i²(p0)δD(p−p0), (1.249) from which it follows that states with negativep0 have negative norms.
The usual remedy is in the redefinition of vacuum. Writing
pµ≡p= (p, ωp) , ωp=|qM2+p2|, (1.250)
Box 1.2: Momentum on mass shell
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Pµ=R TµνdΣν
Tµν = Λ
2(∂µΦ†∂νΦ +∂µΦ∂νΦ†)−Λ
2(∂αΦ†∂αΦ−M2Φ†Φ)δµν
Φ =R dp δ(p2−M2)eipxc(p)
∂µΦ =R dp δ(p2−M2)eipxc(p)ipµ dΣν =nνdΣ
Pµ= Λ 2
Z
dpdp0δ(p2−M2)δ(p02−M2)p0µpνnν
׳c†(p0)c(p) +c(p0)c†(p)´ei(p−p0)xdΣ
δ(p2−M2) =Rds ei(p2−M2)s, dsdΣ = dDx Λ = 1
pp2
ds dτ = 1
Pµ= 12Rdp δ(p2−M2)²(pn)pµ³c†(p)c(p) +c(p)c†(p)´ dp= dωdp p= (ω,p)
ωp=pp2+M2
Pµ= 1 2
Z dp
µωp p
¶ 1 2ωp
³c†(ωp,p)c(ωp,p) +c(ωp,p)c†(ωp,p)
+c†(−ωp,−p)c(−ωp,−p) +c(−ωp,−p)c†(−ωp,−p)´ I. [c(p), c†(p0)] =δD(p−p0) c(p)|0i= 0
1
2ωpc†(ωp,p) =a†(p), 2ω1
pc†(−ωp,−p) =b†(−p)
1
2ωpc(ωp,p) =a(p), 2ω1
pc(−ωp,−p) =b(−p) a(p)|0i= 0, b(−p)|0i= 0
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or
II. [c(p), c†(p0)] =²(p0)δD(p−p0)
1
2ωpc†(ωp,p) =a†(p), 2ω1
pc†(−ωp,−p) =b(p)
1
2ωpc(ωp,p) =a(p), 2ω1
pc(−ωp,−p) =b†(p) a(p)|0i= 0, b(p)|0i= 0
Pµ= 1 2
Z dp
µωp p
¶ ³a†(p)a(p) +a(p)a†(p) +b†(p)b(p) +b(p)b†(p)´
For both definitions I and II the expression for Pµ is the same, but the relations of b(p), b†(p) to c(p) are different.
and denoting p1
2ωpc(p, ωp)≡a(p) , 1
p2ωpc†(p, ωp)≡a†(p), (1.251) p1
2ωpc†(−p,−ωp) , 1
p2ωpc(−p,−ωp) =b†(p), (1.252) let us define the vacuum according to
a(p)|0i= 0 , b(p)|0i= 0. (1.253) If we rewrite the commutation relations (1.244) in terms of the operators a(p),b(p), we find (see Box 1.3)
[a(p), a†(p0)] =δ(p−p0), (1.254)
[b(p), b†(p0)] =δ(p−p0), (1.255) [a(p), b(p0)] = [a†(p), b†(p0)] = [a(p), b†(p0)] = [a†(p), b(p0)] = 0. (1.256)
Box 1.3: Commutation relations
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[c(p), c†(p0)] =αδD(p−p0) α= 1 Case I of Box 1.2
²(p0) Case II of Box 1.2 on mass shell we must insert the δ-function
[c(p), c†(p0)]δ(p2−M2) =αδD(p−p0)
R[c(p), c†(p0)]δ(p2−M2)dp00=αδ(p−p0)
1
2ωp[c(p0,p), c†(ωp,p0)] +2ω1
p[c(p0,p), c†(−ωp,p0)] =αδD(p−p0)
[c(p0,p), c†(−ωp,p0)] = 0, whenp0>0
1
2ωp[c(ωp,p), c†(ωp,p0)] =δ(p−p0)
[c(p0,p), c†(ωp,p0)] = 0, when p0 <0
1
2ωp[c(−ωp,p), c†(−ωp,p0)] =αδ(p−p0)
⇒ [a(p), a†(p0] =δ(p−p0) [b(p), b†(p0] =δ(p−p0)
True either for Case I or Case II with corre-sponding definitions of b(p), b†(p)
Using the operators (1.251), (1.252) and the vacuum definition (1.253) we obtain that the scalar products between the states |pi+ = a†(p)|0i,
|pi−=b†(p)|0i are non-negative:
hp|+|p0i+ = hp|−|p0i− =δ(p−p0),
hp|+|p0i− = hp|−|p0i+ = 0 (1.257) If we now calculate, in the presence of the commutation relations (1.244) and the vacuum definition (1.253), the eigenvalues of the operator P0 we find that they are all positive. Had we used the vacuum definition (1.248) we would have found thatP0 can have negative eigenvalues.
On the other hand, inthe unconstrained theorythe commutation relations (1.182), (1.183) and the vacuum definition (1.248) are valid. Within such a framework the eigenvalues of P0 are also all positive, if the space-like hypersurface Σ is oriented along a positive time-like direction, otherwise they are all negative. The creation operators arec†(p) =c†(p0,p), and they create states |pi =|p0,pi where p0 can be positive or negative. Not only the states with positive, but also the states with negative 0-th component ofmomentum(equal to frequency if units are such that ¯h= 1) have positive energyP0, regardless of the sign ofp0. One has to be careful not to confuse p0 with the energy.
That energy is always positive is clear from the following classical ex-ample. If we have a matter continuum (fluid or dust) then the energy–
momentum is defined as the integral over a hypersurface Σ of the stress–
energy tensor:
Pµ= Z
TµνdΣν. (1.258)
For dust we have Tµν = ρ uµuν, where uµ = dxµ/ds and s is the proper time. The dust energy is thenP0 =R ρ u0uνnνdΣ, where the hypersurface element has been written as dΣν = nνdΣ. Here nν is a time-like vector field orthogonal to the hypersurface, pointing along a positive time-like direction (into the “future”), such that there exists a coordinate system in which nν = (1,0,0,0, ...). Then P0 =R dΣρ u0u0 is obviously positive, even ifu0is negative. If the dust consists of only one massive particle, then the density is singular on the particle worldline:
ρ(x) =m Z
ds δ(x−X(s)), (1.259)
and
P0 = Z
dΣ ds mu0uµnµδ(x−X(s)) =mu0(uµnν).
We see that the point particle energy, defined by means of Tµν, differs from p0 =mu0. The difference is in the factor uµnµ, which in the chosen coordinate system may be plus or minus one.
Conclusion. Relativistic quantum field theory is one of the most suc-cessful physical theories. And yet it is not free from serious conceptual and technical difficulties. This was especially clear to Dirac, who expressed his opinion that quantum field theory, because it gives infinite results, should be taken only as a provisional theory waiting to be replaced by a better theory.
In this section I have challenged some of the cherished basic assumptions which, in my opinion, were among the main stumbling blocks preventing a further real progress in quantum field theory and its relation to gravity.
These assumptions are:
1)Identification of negative frequency with negative energy. When a field is expanded both positive and negative p0 = ¯hω = ω, ¯h = 1, occur. It is then taken for granted that, by the correspondence principle, negative frequencies mean negative energies. A tacit assumption is that the quantity p0 is energy, while actually it is not energy, as it is shown here.
2)Identification of causality violation with propagation along space-like separations. It is widely believed that faster–than–light propagation vi-olates causality. A series of thought experiments with faster–than–light particles has been described [18] and concluded that they all lead to causal loops which are paradoxical. All those experiments are classical and tell us nothing about how the situation would change if quantum mechanics were taken into account10. In Sec. 13.1 I show that quantum mechanics, prop-erly interpreted, eliminates the causality paradoxes of tachyons (and also of worm holes, as already stated by Deutsch). In this section, therefore, I have assumed that amplitudes do not need to vanish at space-like sepa-rations. What I gain is a very elegant, manifestly covariant quantum field theory based on the straightforward commutation relations (1.181)–(1.183) in which fields depend not only on spacetime coordinatesxµ, but also on the Poincar´e invariant parameterτ (see Sec. 1.2). Evolution and causality are related to τ. The coordinate x0 has nothing to do with evolution and causality considerations.
To sum up, we have here a consistent classical and quantum uncon-strained (or “parametrized”) field theory which is manifestly Lorentz co-variant and in which the fields depend on an inco-variant parameter τ. Al-though the quantum states are localized in spacetime there is no problem with negative norm states and unitarity. This is a result of the fact that the commutation relations between the field operators are not quite the
10Usually it is argued that tachyons are even worse in quantum field theory, because their negative energies would have caused vacuum instability. Such an argument is valid in the conventional quantum field theory, but not in its generalization based on the invariant evolution parameterτ.
same as in the conventional relativistic quantum field theory. While in the conventional theory evolution of a state goes along the coordinate x0, and is governed by the componentsP0 of the momentum operator, in the un-constrained theory evolutions goes alongτ and is governed by the covariant Hamiltonian H (eq. (1.161). The commutation relations between the field operators are such that the Heisenberg equation of motion determined by H is equivalent to the field equation which is just the Lorentz covariant Schr¨odinger equation (1.67). Comparison of the parametrized quantum field theory with the conventional relativistic quantum field theory reveals that the expectation values of energy–momentum and charge operator in the states with definite masses are the same for both theories. Only the free field case has been considered here. I expect that inclusion of interactions will be straightforward (as in the non-relativistic quantum field theory), very instructive and will lead to new, experimentally testable predictions.
Such a development waits to be fully worked out, but many partial results have been reported in the literature [19, 20]. Although very interesting, it is beyond the scope of this book.
We have seen that the conventional field theory is obtained from the unconstrained theory if in the latter we take the definite mass fieldsφ(x) and treat negative frequencies differently from positive one. Therefore, strictly speaking, the conventional theory is not a special case of the unconstrained theory. When the latter theory is taken on mass shell then the commutator [φ(x), φ(x0)] assumes the form (1.243) and thus differs from the conventional commutator (1.245) which has the function²(p0) = (1/2)(θ(p0)−θ(−p0)) under the integral.
Later we shall argue that the conventional relativistic quantum field the-ory is a special case, not of the unconstrained point particle, but of the unconstrained string theory, where the considered objects are the time-like strings (i.e. worldlines) moving in spacetime.