Expert: Philip Stahl - 8/11/2006

Question
How does One arrive that a quantum event is plausible when the time interval is 0, and thus the probability and uncertainty of such an event occuring is 0?

Doesn't the uncertainty of quantum mechanics is directly proportional to the time interval allowed for the event; thus arriving at a probability of such an event occuring?

In short... No time = 0 probability.
If T=0, would this not require E=infinite for a quantum event to be plausable?

Likewise, to maintain the probability of a QE ocurring if E=0, would require T=infinite, which is logically impossible to have time exist without a beginning.Isn't it?




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Followup To

Question -
Hello Philip,

In quantum cosmolgy, does absolute nothingness exist? I am told that it does and that even Hawkins ZPE in a vacuum is merely a quantum foam of virtual particles.Absolute nothingness is better understood I am told if I imagine it existing between two objects. The original object becomes/is a singularity point. In nothingness the quantum event occurs resulting in the new object or the universe.

Is my understanding of a quantum fluctuation in absolute nothingness being the trigger to BB causality correct?

Answer -
Hello.

Okay, you actually have several issues that you are looking to address.

First, it is difficult to say with certainty that "absolute nothingness" exists - because even in quantum cosmology we only go down to a scale equal to the Planck length. That is: L_P = 1.6 x 10^ -33 cm. This is the lowest limit at which the operational definitions work. Obviously, if there is something of scale L << L_P we wouldn't know of it, and arguing that there is 'absolute nothingness' at L_P totally leaves open the question of what exists (or not) at L!

The notion of quantum "foam" and fluctuations within it was perhaps first explicated by John Wheeler in the early 70s (e.g. in his essay, 'Is Physics Legislated by Cosmogony?'). The basic notion is that - given some region of observed dimension L, a calculated quantum field fluctuation will be of order:

delta (f) ~ (hc)^1/2/ L^2

where h is Planck's constant (h = 6.62 x 10^-34 J*s) and c the speed of light.

Quantum fluctuations in normal metrics are of order:

delta (g) ~  L_P/ L

where the numerator is the earlier defined 'Planck length'.

Wheeler calculated the effective mass-energy density of vacuum fluctuations, down to Planck length as ~ 10^94 g/cm^3

This is probably what led the late quantum physicist David Bohm to write ('Wholeness and the Implicate Order', p. 191):


"What we call empty space contains an immense background of energy and matter as we know it is a small quantized excitation on top of this background, rather like a tiny ripple on a vast sea... this vast (energy) sea may play a part in understanding the cosmos as a whole"

The problem is that no one knows how to actually get at this energy and use it in any meaningful way.

Using "singularity points" or singularities in any operational definition becomes problematical - because the whole thrust of quantum field theory is to *avoid them*! Thus, physics abhors singularities or "divergences" as they are known.

In typical QFT texts one can find whole chapters devoted to the procedure we have come to know as "renormalization" - the entire objective of which is to "remove divergences".  This usually involves some re-interpretation (especially mathematical!) of any number of parameters and components to the problem.

In this light, employing any singularity as part of a definition of quantum nothingness, is not likely to lead anywhere.

Lastly, it is true that some physicists have proposed a quantum fluctation to account for the Big Bang. For example, T. Padmanabhan, ‘Universe Before Planck Time – A Quantum Gravity Model, (in Physical Review D, Vol. 28, No. 4, p. 756.) attributes the instantaneous formation of the universe via a possible quantum fluctuation. To do this he treats the conformal part of space-time as a quantum variable.

Without going into all the complex mathematics entailed, Padmanabhan employed integrals related to the “action” (J) as a function of time, t. He proceeded by solving for the expansion factor S(t) using two separate energy equations, one of which (2.15 in his paper) bears a remarkable resemblance to the basic time-dependent Schrodinger quantum wave equation. The potential energy term he uses is remarkably similar to that for a quantum harmonic oscillator.

The most masterful section in his paper is III. ‘Geometry of the Quantum Universe’ wherein spacetime itself is taken to be in a particular quantum state U(q, t). He then assumes “stationary states” (given by the variable Q) for the early universe that are independent of time and for which all the dynamics “are contained in S(t).

The conformal factor Padmanabhan uses is 'alpha', which is a purely quantum mechanical parameter, defined from his equation (2.24):

alpha = S(t)^ 6  (omega(t))^2

Physically, it is found that the conformal factor (alpha) contributes a negative energy density.The point is, the metric and treatment is feasible since non-trivial and matterless solutions exist. Thus, the cosmos can be incepted and expands because of the negative-energy density of the conformal factor.

It is also this basis that provides the model for the instantaneous formation of the universe by a possible quantum fluctuation that arises when a particular threshold is crossed near alpha = 0 (from quantum to classical domains).  As Padmanbhan shows in his paper, such a cosmos from “nothing” is perfectly justified in the context of the model, and follows from the basis of the equations, the light cone, scale factor restrictions and so on.

This means the limits at spontaneous inception must definitely be for *acausal determinism*, not classical causality. This is a crucial distinction that I cannot emphasize enough.

The concept of vacuum energy associated with a negative energy density, can be embodied in a kind of simplified equation of state, viz.:

w = (Pressure/ energy density) = -1

This is consistent with Einstein's general theory of relativity - which one could say approaches the status of a 'basic law of physics'.  In this case, the existence of a negative pressure is consistent with general relativity's allowance for a "repulsive gravity" - since any negative pressure has associated with it gravity that repels rather than attracts. (See, e.g. 'Supernovae, Dark Energy and the Accelerating Universe', by Saul Perlmutter, in Physics Today, April, 2003, p. 53.)

Of course, simple algebra applied to the above also shows that the energy density would have to be negative, e.g. energy density = - (pressure). Thus, we see that Padmanabhan’s “negative energy density” referenced earlier is really a form of dark energy!

Interestingly, this is also what astronomers now attribute as the primary cause for accelerating the expansion rate.

In the end, this may very well be the only practical format in which vacuum energy ever appears.


Answer
Hello.

The problem I've usually found dealing with issues such as quantum cosmology, or even cosmic inflation - is that questions can be asked in a regular, common sense sort of fashion - but the subset of the universal set of answers most appropriate to really address them has so much detail that they won't be understood in the context.

In other words, to make a long story short, it is quite possible to ask reasonable sounding questions for which practical and 'normal'(not to mention reasonably tractable) answers are impossible. This is a case in point - since the really detailed and exact response would necessitate delving into things that are truly beyond the scope of an 'All experts' type response.

That being said, let me try to provide a more or less reasonable and tractable answer.

In his definitive paper, ‘Universe Before Planck Time – A Quantum Gravity Model, in Physical Review D, Vol. 28, No. 4, p. 756, T. Padmanabhan uses as a time coordinate *hyperboloids of constant distance*, inside the light cone of a point in what we call 'de Sitter space'. The point itself, and its light cone, are the Big Bang of the Friedmann model, where the scale factor goes to zero.

But they are not singular. By which I mean, as per the previous response, they don't have any infinite divergences.

Instead, the spacetime *continues through the light cone to a region beyond*. It is this region that deserves the name, the pre -Big Bang scenario.  The trouble is trying to explicate it- since the underpinning concept is largely a mathematical one that depends upon constructing certain complex mathematical expressions called 'Lagrangians'. I don't want to get into that - for one thing- because I've no remote idea what your math background is.

It is also this feature that provides a basis for the model for the instantaneous formation of the universe by a possible quantum fluctuation that arises when one treats the conformal part of space-time as a quantum variable.

As Padmanbhan shows in his paper, such a cosmos from nothing is perfectly expected and indeed, "follows from the basis of the tensor set up, the light cone restrictions and so on." As he notes, if the Euclidean four-sphere were perfectly round, both the closed and open analytical continuations (using complex integrals around specific paths), would inflate forever.

This would mean they would never form galaxies. A perfect round four sphere has a lower action, and hence a higher a-priori probability than any other four -metric (x, y, z, t) of the same volume.

At a less complex level, though further removed from the more exact details, consider the energy-time uncertainty principle:

(delta E)  (delta T) ~ h-bar

where h-bar = h/ 2 pi

where h is the Planck constant (= 6.62 x 10^-34 J-s)

If any time interval is 0, then delta T  = 0

and delta E (change in energy) =   h-bar / 0  =  oo


Clearly then, this case is impossible since it yields a cosmos with infinite energy - when we know the total mass-energy had to be *finite*/ Thus, one cannot have a case or instance wherein delta T = 0.

At the same time, it is seen that any uncertainty in the delta T can lead to a delta E (change in energy) that can be the amount of energy needed for the inception of the cosmos. This will, of course, depend on the exact value of delta T.

As you also quite correctly pointed out, making delta E = 0 yields infinite time which is also not feasible.

What we are left with is that both uncertainties can never go to zero - BUT this is not the same as asserting that at some time t = 0 all *values of physical quantities go to zero*.

As I already noted, the use of hyperboloids of constant distance, *inside the light cone*  of a point in de Sitter space- eliminates any such time zero problem of this form.

One last point, since the probability function (pardon the deformed integral sign!)

P =  

**
*  U(t) U*(t) dt
**


with U(t) = exp (iEt/h)

U*(t) = exp (-iEt/h)

with the conditions noted earlier, then P must be non zero. In other words, there exists a finite probability that the universe can spontaneously be incepted via a quantum fluctuation.