Boost Coordinate Frame as a Valid Coordinate Frame in Quantum Field Theory

The consistent use of geometrical clocks puts constraints on the mathematical formulation of waves propagating in the inertially expanding coordinate frame $F$. In this frame, a standard inertially expanding clock AB characterized by Doppler frequency shift factor Eq.(25),

$$k_{AB}=e^{(\tau_B-\tau_A)}\equiv e^{\Delta\tau},$$

generates pulses whose null histories as depicted in Figure 2 are

$$\xi e^{\tau} = be^{n_2\Delta\tau}~~~~~n_2=0,\pm 1, \cdots~~$$ (26)

$$\xi e^{-\tau} = be^{n_1\Delta\tau}~~~~~n_1=0,\pm 1, \cdots~~.$$ (27)

The graduation events calibrated by this geometrical clock yield therefore the following discrete boost coordinates

$$\xi = be^{N\Delta\tau}, ~~~~~N=\frac{n_2+n_1}{2}$$ (28)

$$e^\tau = e^{M\Delta\tau}, ~~~~~M=\frac{n_2-n_1}{2}~.$$ (29)

As illustrated in Figure 9 and discussed in Section V+.1667emB, they are the boost coordinates of the $M$th identically constructed clock with its $N$th ticking event.

In a paper some time ago Padmanabhan (1990) Padmanabhan considered the evolution of normal modes of the wave equation $(\Box -m^2)\psi=0$ in the boot-invariant coordinate frame $F$.

Starting with a normal mode characterized by positive boost frequency in the distant past of $F$, he observed that this mode, in compliance with the wave equation, evolved into a mixture of positive and negative frequencies in the distant future of $F$. From the viewpoint of quantum theory such a mixture indicates a production of particles and antiparticles. This formulation of waves propagating in $F$ therefore leads to the mathematical prediction that, in analogy with Parker's particle-antiparticle creation mechanism Parker (1982) due to a time-dependent gravitational field, particles and antiparticles get created because of the time-dependence of the boost-invariant metric, Eq.(33), in $F$.

This prediction is, of course, invalid. It contradicts the absence of any such particle creation in flat spacetime, where there is no gravitational field. But the procedure leading to this contradiction, Padmanbhan points out, is mathematically sound and completely conventional [our emphasis]. In order to avoid this contradiction he proposes that, within the context of quantum theory (i.e. particle-antiparticle production), one exclude Bondi and Rindler's spacetime coordinatization as physically inadmissible.

However happy one must be about the scrutiny to which that coordinatization has been subjected, one must not forget that Padmanabhan's procedure leading to to the above contradiction is far from ``completely conventional''. In fact, it violates the central principle of measurement (Section III): ``once a standard of time has been chosen, it becomes immutable for all subsequent measurements''. Here is how the violation occurs:

In spacetime sector $F$, where the invariant interval has the form

$$ds^2=-d\xi^2+\xi^2d\tau^2+dy^2+dx^2~,$$ (30)

the normal modes of the wave equation $(\Box -m^2)\psi=0$ have the form

$$\psi=\psi_k(\xi,\tau)e^{i(k_yy+k_x x)}~,$$

Where $\psi_k(\xi,\tau)$ satisfies

$$\left[ \frac{1}{\xi}\frac{\partial}{\partial\xi} \xi \frac{\... ...rac{\partial^2}{\partial\tau^2}+k^2 \right] \psi_k(\xi,\tau)=0$$

with $k^2\equiv k_y^2 +k_x^2 +m^2$. A typical normal mode has the form

$$\psi = J_{-i\omega}(k\xi)e^{i\omega\tau} e^{i(k_yy+k_x x)}$$ (31)

$$= \frac{1}{2} \left[ H_{-i\omega}^{(1)}(k\xi)+H_{-i\omega}^{(2)}(k\xi) \right] e^{i\omega\tau} e^{i(k_yy+k_x x)}$$ (32)

Measuring its field consists of sampling it at the events (time $\xi$ and location $\tau$) controlled and calibrated by a set of identically constructed clocks. If these clocks are inertially expanding clocks as in Figure 9, then the sampling events are given by Eqs.(31)-(32), and the sampled field values are

$$\begin{eqnarray*} \psi &=&J_{-i\omega}(kb e^{N\Delta\tau}) e^{i\omega M\Delta\ta... ...^{i\omega M\Delta\tau}e^{i(k_yy+k_x x)}\textrm{ as }N\to -\infty \end{eqnarray*}$$

If the samples are are close enough (i.e. $\Delta\tau$ small enough), then, using Shannon's sampling theorem, one reconstructs the field from the sampled values of its field.

Note that even though this clock-controlled sampling measurement reconstructs the the field uniquely in the distant past ($N\to -\infty$) of $F$, it is clear that this is not the case in the distant future ($N\to \infty$). Regardless how small one makes the separation between the sampling events in the asymptotic past, in the asymptotic future the inertially expanding clocks tick at such a slow rate (compared to any atomic clock) that there is no possibility of reconstructing the field from the sampling measurements. Indeed, in the distant future ( $\xi=be^{N\Delta\tau}\to\infty$), the field, Eq.(35),

$$\begin{eqnarray*} \psi\approx\frac{1}{2}\sqrt{\frac{2}{\pi k\xi}} &~&\!\!\!\!\!\... ...\pi/4)} \right] \\ &~&\times ~e^{i\omega \tau}e^{i(k_yy+k_x x)} \end{eqnarray*}$$

oscillates at a steady rate as a function of (atomic=proper) $\xi$. But the sampling events, as one can see readily from Figure 9, are so sparsely spaced as $N\to \infty$ that there is more than one oscillation between them. Consequently, reconstruction becomes non-unique and hence out of the question. In particular, sampling measurements controlled by an inertially expanding clock are incapable of distinguishing normal modes traveling into opposite directions, to say nothing about identifying their oscillation frequencies in the distant future of $F$.

Would an atomic clock do better? The answer is yes. But only for sampling measurements made in the distant future ( $\xi=be^{N\Delta\tau}\to\infty$). For the distant past ( $N\to\infty,~\xi=be^{N\Delta\tau}\to 0$) atomic clocks are just as useless as inertially expanding clocks are for the distant future: the clocks simply do not sample the field fast enough to identify its boost oscillation frequency.

Thus neither atomic clocks nor inertially expanding clocks can give measurements which identify the nature of the field in both the asymptotic past and the asymptotic future of $F$. One can measure the field in one or the other but not both.

A claim that in boost-invariant sector $F$ a pure positive boost frequency ($\omega$) mode evolves into a superposition of positive and negative inertial frequency ($\pm k$) modes is wrong. This is because it makes the tacit assumption that one change inertially expanding to static atomic clocks in midstream. Making such a change would go counter to the central principle of measurement (Section III): ``once a standard has been chosen it becomes immutable for all subsequent measurements''. Violating it would make a standard into a non-standard.

But a standard is precisely what is needed, otherwise there would be no way of assigning a frequency and a direction of propagation to normal modes, the key ingredients to mode amplification and hence to particle creation as formulated in quantum field theory. Put differently, an assertion that a mode having a positive frequency evolve mathematically into a mixture of positive and negative frequency modes must be accompanied by a specification of a (system of commensurable) standard clock(s).

It is evident that in sector $F$ no such standard exists. Consequently, one is not entitled to claim that mathematical analysis of free fields in that sector predicts the creation of particles.