Before inflation ?


by big_bounce
Tags: inflation
big_bounce
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Feb16-13, 08:54 AM
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Hello all .

I have two questions :

1 - Before inflation elementary particles such as matter and photons could not be created and destroyed or could be created and destroyed but could not be permanent ?
I mean can we say matter existed before inflation but it was created and destroyed in frame time ?


2 - if elementary particles and matter came from energy of vacuum , can we say energy has momentum ? because matter has momentum ? ( refer to conversion of momentum )
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Naty1
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Feb16-13, 12:58 PM
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Before inflation elementary particles such as matter and photons could not be created and destroyed or could be created and destroyed but could not be permanent ?
Nobody knows what existed before the big bang....Maybe the big bang was a 'one time event' [the most popular theory] or maybe we inhabit one or many universes...or maybe we inhabit a cyclic universe. I don't think anything is 'permanent'....but maybe something is.

- if elementary particles and matter came from energy of vacuum , can we say energy has momentum ? because matter has momentum ? ( refer to conversion of momentum
That's one way to think about it. An easy way to remember that 'energy has momentum' is to recall that 'photons have momentum'....Photons are the local excitations [quanta] of the electromagnetic field. [So a 'particle' does not have to have mass to have energy or momentum.] We know the electromagnetic field carries energy because radios, alternators in our cars, and transformers, for example, really work!


In empty space, the photon moves at the speed of light, c. Its energy E and momentum p are related by E = pc. There is an easy to understand derivation and description here:

http://en.wikipedia.org/wiki/Photon_...cal_properties
big_bounce
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Feb16-13, 04:39 PM
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Quote Quote by Naty1 View Post
Nobody knows what existed before the big bang....Maybe the big bang was a 'one time event' [the most popular theory] or maybe we inhabit one or many universes...or maybe we inhabit a cyclic universe. I don't think anything is 'permanent'....but maybe something is.
I was not talking about big bang .
I read , big bang is initial expansion from singularity and from 0s up to 1 Planck time called quantum gravity era not big bang . I just read and not sure

Anyway , i meant about "Before inflation" is the GUT epoch not before the big bang .
In this era elementary particles such as matter and photons could not be created and destroyed or could be created and destroyed but could not be permanent and break down to energy fast ?

Quote Quote by Naty1 View Post
That's one way to think about it. An easy way to remember that 'energy has momentum' is to recall that 'photons have momentum'
But photons are not energy ( or pure energy ) they are elementary particles . elementary particles and energy are not same thing .

Quote Quote by Naty1 View Post
In empty space, the photon moves at the speed of light, c. Its energy E and momentum p are related by E = pc. There is an easy to understand derivation and description here
I agree with you that Energy is related with momentum but can we say momentum comes from energy ? and in big bang theory momentum came from vacuum energy ?

Naty1
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Feb16-13, 07:57 PM
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Before inflation ?


In this era elementary particles such as matter and photons could not be created and destroyed or could be created and destroyed but could not be permanent and break down to energy fast ?
Try reading the first section here.....it's a good introduction in half a dozen or so paragraphs.

http://en.wikipedia.org/wiki/Chronology_of_the_universe

At first, temperature and energy was too high for particles as we know them today to form. So most particles came from the following period of inflationary expansion...some from early, many more from during the expansion. I do not what existed before inflation except for sure energy; I do not know its form.

But photons are not energy ( or pure energy ) they are elementary particles . elementary particles and energy are not same thing .
yes, same thing... photons are localized energy fields, QUANTA. E= hf describes the energy of a photon. Has no mass. All particles can be seen as localized energy fields; some have mass, a few don't.

I agree with you that Energy is related with momentum but can we say momentum comes from energy ? and in big bang theory momentum came from vacuum energy ?
this is probably close...but best to say energy and momentum are closely related.
standard inflation likely was sourced from some sort of vacuum energy. But the cosmological model we have adds to that a high energy background vacuum to power inflationary expansion, called slow roll inflation. I've seen that add on energy field described as a HIGGS field and in some models it takes several types of HIGGS fields, like one for each particle. It's a manual adjustment conceived by Alan Guth [and others] to mimic observations we can make today. And so is the shape of the field....it has to be shut off after a brief period.

All energy and momentum, and gravity,and particles came from the big bang via inflationary expansion. As we get closer and closer to the time of the big bang, both general relativity and quantum mechanics, most recently quantum gravity, breaks down so I am unsure if anybody knows what broke loose first...that is, what order did space,time,energy,gravity appear as we know them today? Which 'caused' which?

At the moment of the big bang it appears everything was unified in a very high energy, very unstable environment. 'Spontaneous symmetry breaking' lead to the diffusion of all that initial energy into all the forms of matter and energy we see today. I guess it's possible there is something between inflation and the big bang we don't yet know.

It turns out all MATTER particles up to LEAD came from the initial big bang; atoms heavier than lead are born in supernovas...when they explode those heaver particles are spread in galaxies and are included, for example, in the composition of earth and our moon.
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Feb16-13, 10:07 PM
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Quote Quote by Naty1 View Post
At first, temperature and energy was too high for particles as we know them today to form. So most particles came from the following period of inflationary expansion...some from early, many more from during the expansion. I do not what existed before inflation except for sure energy; I do not know its form.
To my knowledge the temperature was so high that particles and antiparticles were continually created and annihilated over and over again. This only stopped once the universe cooled down to the point that the creation of various particles was no longer possible.

It turns out all MATTER particles up to LEAD came from the initial big bang; atoms heavier than lead are born in supernovas...when they explode those heaver particles are spread in galaxies and are included, for example, in the composition of earth and our moon.
I don't there was any significant creation of atoms beyond Helium other than a tiny tiny bit of Lithium. Everything else up to Iron has been created inside stars, and everything beyond that has been created in supernovas.
big_bounce
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Feb17-13, 04:26 AM
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Quote Quote by Naty1 View Post
yes, same thing... photons are localized energy fields, QUANTA. E= hf describes the energy of a photon. Has no mass. All particles can be seen as localized energy fields; some have mass, a few don't.
About photons and energy are same things , i don't agree with you by refer to these posts which i will link below :
http://www.physicsforums.com/showpos...36&postcount=6
http://www.physicsforums.com/showpos...28&postcount=2


Moreover , what's energy fields ? i didn't find any concept and define about energy fields in physics .
Photons are forms of electromagnetism radiation which made of two fields ( The electric field is in a vertical plane and the magnetic field in a horizontal plane ) and these fields carry energy and can possess momentum and energy , not same to energy .

Quote Quote by Drakkith View Post
To my knowledge the temperature was so high that particles and antiparticles were continually created and annihilated over and over again.
Thanks i more agree with this but i saw a figure in Stanford university's website that was shown particles and photons were existed after inflation . the figure was same this
So pair production and create particle is meaningless before end of inflation because photons were not existed .
http://universe-review.ca/I02-07-cosmology.jpg
Naty1
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Feb17-13, 08:07 AM
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About photons and energy are same things , i don't agree with you by refer to these posts which i will link below :
http://www.physicsforums.com/showpos...36&postcount=6
http://www.physicsforums.com/showpos...28&postcount=2
Those posts provide more details...details I with which I agree. I did not know your level of understanding so I keep things simple at first.

It turns out all MATTER particles up to LEAD came from the initial big bang; atoms heavier than lead are born in supernovas...when they explode those heaver particles are spread in galaxies and are included, for example, in the composition of earth and our moon.
I don't there was any significant creation of atoms beyond Helium other than a tiny tiny bit of Lithium. Everything else up to Iron has been created inside stars, and everything beyond that has been created in supernovas.
oops, I keep doing that...posting LEAD instead of IRON. When I say all matter up to iron came from the big bang it means that matter constituents became available for stars to form
and from those nuclear reactions come all the remaining elements up to iron. The DIRECT particles from the inflationary era appear to be CMBR photons and neutrinos and these are believed to be the most abundant particles in the Universe.


what's energy fields ? i didn't find any concept and define about energy fields in physics .
There are many different perspectives on FIELDS:

Here are a few insights:
Carlo Rovelli
A particle is in some sense the smallest volume/unit in which the field or action of interest can operate….Most discussions regarding particles are contaminated with classical ideas of particles and how to rescue these ideas on the quantum level. Unfortunately this is hopeless....., uniquely-defined particle states do not exist in general, in QFT on a curved spacetime....... A particle detector measures a local observable field quantity (for instance the energy of the field, or of a field component, in some region). This observable quantity is represented by an operator that in general has discrete spectrum. The particles observed by the detector are the quanta of this local operator..... , strictly speaking there are two distinct notions of particles in QFT. Local particle states correspond to the real objects observed by finite size detectors. .... On the other hand, global particle states....can be defined only under certain conditions.
http://arxiv.org/abs/gr-qc/0409054

Marcus did not like some of the above description:
....This means that particles (not some mysterious fields) are the objects studied by real experimental physics. If "curved spacetime" does not agree with the particle concept, so bad for the "curved spacetime".
So locally the particle concept seems well defined. The problem arises when you want to make statements which are globally valid, or when you change the reference frame as you do in the Unruh effect. No one knows what space is,,,,, any more than they know what time is...or mass, energy, gravity,nor dark energy nor FIELDS.......for most we can describe the observational effects but not the fundamental origin, nor "what is..."

I think I got this one from Wikipedia:
...There is not a definite line differentiating virtual particles from real particles — the equations of physics just describe particles (which includes both equally). The amplitude that a virtual particle exists interferes with the amplitude for its non-existence; whereas for a real particle the cases of existence and non-existence cease to be coherent with each other and do not interfere any more. In the quantum field theory view, "real particles" are viewed as being detectable excitations of underlying quantum fields. As such, virtual particles are also excitations of the underlying fields, but are detectable only as forces but not particles.
TomStoer posted this which I liked:

Particles appear in rare situations, namely when they are registered.
and a related view I saved from somewhere:

Matter is that which has localized mass-energy, while space-time does not.
Is space-time composed of fields?? Is matter [particles]? ....good subjects for more debates...

Regarding the big bang: quantum fluctuations in the inflationary vacuum become quanta [particles] at super horizon scales. Particle production via changing gravitational fields and expansion is believed a real phenomenon. It seems that expansion of geometry itself, especially inflation, can produce matter. This implies fields are the fundamental constitutents of particles....localized quanta with particular characteristics following conservation laws.

Other theoretical examples where geometric circumstances create real (not virtual) particles are Hawking radiation at a BH horizon and Unruh radiation caused by an accelerating observer. This viewpoint is quite different, for example, from that expressed by Marcus.

We still have lots more to learn!
Mordred
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Feb17-13, 10:39 AM
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There is a couple other virtual particle to real particle creation processes. One being Parker radiation. This one being due to expansion or contraction of spacetime. Unlike Hawking or Unruh radiation. Parker Radiation does not require a horizon. All of these mentioned are forms of blackbody radiation. The main difference is the perturbation source.
I've always wondered if these processes can be condensed into one process model including the Casimer effect? May make an interesting thread lol

Ive always thought of spacetime as a geometric field makes certain concepts easier to relate to. However thats a personal opinion, that may or may not be accurate
Here is a paper on Parker.
http://arxiv.org/abs/1203.1173
Drakkith
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Feb17-13, 11:35 AM
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Quote Quote by big_bounce View Post
Thanks i more agree with this but i saw a figure in Stanford university's website that was shown particles and photons were existed after inflation . the figure was same this
So pair production and create particle is meaningless before end of inflation because photons were not existed .
http://universe-review.ca/I02-07-cosmology.jpg
First and foremost, we have absolutely no evidence either way. However it is believed that there was indeed matter and photons before/during inflation. And just because one picture somewhere doesn't show matter or photons before inflation doesn't mean it's the absolute truth. In fact your link looks like it has a squiggly photon line before inflation.

From wiki: http://en.wikipedia.org/wiki/Inflationary_epoch

It is not known exactly when the inflationary epoch ended, but it is thought to have been between 10−33 and 10−32 seconds after the Big Bang. The rapid expansion of space meant that elementary particles remaining from the grand unification epoch were now distributed very thinly across the universe. However, the huge potential energy of the inflation field was released at the end of the inflationary epoch, repopulating the universe with a dense, hot mixture of quarks, anti-quarks and gluons as it entered the electroweak epoch.


If we have charged particles interacting, then we have EM radiation and photons. Note that the matter from the grand unification epoch was only spread very thinly AFTER inflation ends, not before, so it could have had plenty of energy, density, and temperature to have particle-antiparticle creation and annihilation.
Mordred
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Feb17-13, 11:46 AM
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The particle production process of elements is best described by BBN. Big bang nucleothynthesis. One of my favourite sites has a decent write up.

http://www.astro.ucla.edu/~wright/BBNS.html
Naty1
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Feb17-13, 03:27 PM
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bigbounce...I came across this brief description which relates to why it is not so good to claim momentum comes from energy.....


..Energy and momentum are not entirely separate entities and one observer's energy gets split into another observer's momentum in the same way as space and time...
This is not so easy to understand, but Richard Feynman in his book SIX NOT SO EASY PIECES says that replacing in the Lorentz transformations x with px (for momentum) and replacing t with E (for energy as mc2] yields the four vector momentum.....
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Feb20-13, 04:26 PM
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Quote Quote by Drakkith View Post
If we have charged particles interacting, then we have EM radiation and photons. Note that the matter from the grand unification epoch was only spread very thinly AFTER inflation ends, not before, so it could have had plenty of energy, density, and temperature to have particle-antiparticle creation and annihilation.

I read an article on Wikipedia which i will link below

During the grand unification epoch, physical characteristics such as mass, charge, flavour and colour charge were meaningless.
http://en.wikipedia.org/wiki/Grand_unification_epoch

So if charge and mass are meaningless in GUT epoch , creation of matter is meaningless in this time because matter has mass and charge , like electron .
Is that correct ?
Drakkith
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Feb20-13, 04:47 PM
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Quote Quote by big_bounce View Post
So if charge and mass are meaningless in GUT epoch , creation of matter is meaningless in this time because matter has mass and charge , like electron .
Is that correct ?
A good question that I'm afraid I cannot answer. It would be interesting to hear from someone more knowledgeable than I am whether or not the different forces and subsequent particle properties mean anything when the forces are unified.
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During this period, three of the four fundamental interactions—electromagnetism, the strong interaction, and the weak interaction—were unified as the electronuclear force. Gravity had separated from the electronuclear force at the end of the Planck era. During the grand unification epoch, physical characteristics such as mass, charge, flavour and colour charge were meaningless. The grand unification epoch ended at approximately 10−36 seconds after the Big Bang. At this point several key events took place. The strong force separated from the other fundamental forces.....
http://en.wikipedia.org/wiki/Grand_unification_epoch



'meaningless" is a really,really poor term....'indistinguishable' or perhaps 'undefined' or "mathematically unclear" would be much better. In the Planck era just prior, gravity was also 'meaningless' and yet it popped out....then out popped the strong force...these became a unique entity at lower energies and survives still.

creation of matter is meaningless in this time ....
that IS the general idea, but much better to say I think mass had not yet emerged from the extremely high energy state of the early universe...all that existed at the time is energy....and other stuff we might not recognize, like fluctuating topology a midst all the quantum energy.

It's analogous, I think, to saying space and time are 'meaningless' at Planck scale...lost amidst quantum foam. Just because we don't understand that environment doesn't mean it's hidden constituents are 'meaningless'


[I personally take these explanations with a grain of salt, as the best we can do now....but quantum gravity will hopefully provide the real insights...when the four forces are hopefully unified at or near the Planck scale that preceded the GUT era. ]
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Feb20-13, 05:28 PM
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Quote Quote by Drakkith View Post
A good question that I'm afraid I cannot answer. It would be interesting to hear from someone more knowledgeable than I am whether or not the different forces and subsequent particle properties mean anything when the forces are unified.
The only decent expression I've seen when all the forces, energy is combined is that the energy potential = zero. I believe this is one of the goals of supersymmetry. However I've had great difficulty finding reliable sources on the subject. I too wait on the a resolving of GUT lol.

edit my mistake I described the planch epoch, the GUT epoch gravity is seperated. This era is sometimes referred to as the electrnuclear epoch.

I found a half decent breakdown of the various epochs


the statement on energy density is described as infinite in this paper, I beleive it was a paperin regards of a universe from nothing scenario that had the energy density at zero, sorry for that confusion.

http://www.nicadd.niu.edu/~bterzic/P...Lecture_13.pdf
Naty1
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Lisa Randall [Harvard physics professor] in WARPED PASSAGES has an interesting no mathematical explanation in Chapters 10,11..."ORIGIN OF ELEMENTARY PARTICLES>>>>>
and SCALING AND GRAND UNIFICATION...

I'm pretty sure she is explaining quantum mechanic interactions as understood in the Standard Model. One starts with symmetry [uniformity] in a very high energy unstable environment...only when lower energies are reached can a Higgs field emerge based on initial spontaneous symmetry breaking and do "...the elementary particles of the Standard Model acquire mass,,.......the vacuum carries weak charge, it does not carry electric charge.......the dependence on energy and distance is over and above the classical separation dependence of forces....." and on and on....

Frankly, she has more detail than I need to know....
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Quote Quote by big_bounce View Post
Hello all .

I have two questions :

1 - Before inflation elementary particles such as matter and photons could not be created and destroyed or could be created and destroyed but could not be permanent ?
I mean can we say matter existed before inflation but it was created and destroyed in frame time ?


2 - if elementary particles and matter came from energy of vacuum , can we say energy has momentum ? because matter has momentum ? ( refer to conversion of momentum )
If you want answers to your question you must specify which MODEL of the pre-inflation universe. The answers depend on which theoretical model you use, of the universe around the start of expansion. The models have not been tested enough yet for one to have gained acceptance over the others.

If you want answer according to the LQC "Big Bounce" model then it is not too hard to say something because there are some fairly clear ideas proposed about matter and geometry around start of expansion.

Here is one LQC approach that appeared recently:
http://arxiv.org/abs/1211.6269
The Matter Bounce Scenario in Loop Quantum Cosmology
Edward Wilson-Ewing
(Submitted on 27 Nov 2012)
In the matter bounce scenario, a dust-dominated contracting space-time generates scale-invariant perturbations that, assuming a nonsingular bouncing cosmology, propagate to the expanding branch and set appropriate initial conditions for the radiation-dominated era. Since this scenario depends on the presence of a bounce, it seems appropriate to consider it in the context of loop quantum cosmology where a bouncing universe naturally arises. It turns out that quantum gravity effects play an important role beyond simply providing the bounce. Indeed, quantum gravity corrections to the Mukhanov-Sasaki equations significantly modify some of the results obtained in a purely classical setting: while the predicted spectra of scalar and tensor perturbations are both almost scale-invariant with identical small red tilts in agreement with previous results, the tensor to scalar ratio is now expected to be r ≈ 10-4, which is much smaller than the original classical prediction. Finally, for the predicted amplitude of the scalar perturbations to agree with observations, the critical density in loop quantum cosmology must be of the order 10-9ρPl .
8 pages

Here is another even more recent LQC paper on the topic you were asking about:
http://arxiv.org/abs/1302.0254
The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations
Ivan Agullo, Abhay Ashtekar, William Nelson
(Submitted on 1 Feb 2013)
Using techniques from loop quantum gravity, the standard theory of cosmological perturbations was recently generalized to encompass the Planck era. We now apply this framework to explore pre-inflationary dynamics. The framework enables us to isolate and resolve the true trans-Planckian difficulties, with interesting lessons both for theory and observations. Specifically, for a large class of initial conditions at the bounce, we are led to a self consistent extension of the inflationary paradigm over the 11 orders of magnitude in density and curvature, from the big bounce to the onset of slow roll. In addition, for a narrow window of initial conditions, there are departures from the standard paradigm, with novel effects ---such as a modification of the consistency relation between the ratio of the tensor to scalar power spectrum and the tensor spectral index, as well as a new source for non-Gaussianities--- which could extend the reach of cosmological observations to the deep Planck regime of the early universe.
64 pages, 15 figures
========================
THIS IS IMPRACTICALLY LONG FOR OUR PURPOSES. I would suggest only sampling this excerpt on page 53:
==quote page 53==
o limit our numerical simulations to φB 􏰭 2 is not physically restrictive.
To summarize, by analyzing the pre-inflationary dynamics in detail we arrived at two main conclusions. First, there do exist natural initial conditions at the bounce which lead to a completion of the standard inflationary scenario to include the quantum gravity regime. In this completed theory, one has a consistent evolution all the way from the deep Planck regime that accounts for the inhomogeneities seen in the CMB. Since the origin of the large scale structure can be traced back to these inhomogeneities, now one can systematically trace back the seeds of this structure to the quantum fluctuations of the initial state at the LQC bounce itself. Second, there is a narrow window in the φB parameter space for which the state at the onset of inflation would not be the BD vacuum. While the LQC and the standard inflation predictions are both compatible with current observations, future observations should be able to distinguish between the two. Thus, there is a potential to extend the reach of observational cosmology all the way to the Planck scale. Of course, since the window is narrow, the Ďa priorií probability of its being realized in Nature is small. This is compensated by the fact that, if observations are compatible with φB being in this window, the initial conditions would be narrowed down tremendously, making very detailed calculations and predictions feasible.
==endquote==
BTW "BD vacuum" means "Bunch-Davies vacuum"---you could probably look that up in Wikipedia.
We have to find something that is more to the point.
Let's glance at this:
http://arxiv.org/abs/1211.1354
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era
Ivan Agullo, Abhay Ashtekar, William Nelson
(Submitted on 6 Nov 2012 (v1), last revised 16 Jan 2013 (this version, v2))
Cosmological perturbations are generally described by quantum fields on (curved but) classical space-times. While this strategy has a large domain of validity, it can not be justified in the quantum gravity era where curvature and matter densities are of Planck scale. Using techniques from loop quantum gravity, the standard theory of cosmological perturbations is extended to overcome this limitation. The new framework sharpens conceptual issues by distinguishing between the true and apparent trans-Planckian difficulties and provides sufficient conditions under which the true difficulties can be overcome within a quantum gravity theory. In a companion paper, this framework is applied to the standard inflationary model, with interesting implications to theory as well as observations.
50 pages, published in Physical Review D.

The Planck Era is the era around the bounce, which came before inflation, so this might have some answers to your questions.
marcus
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Feb20-13, 08:37 PM
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Continuing with http://arxiv.org/abs/1211.1354
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era
Ivan Agullo, Abhay Ashtekar, William Nelson

Well this gives only a limited partial satisfaction. They introduce a kind of general Quantum Field Theory (QFT) sort of matter that participates in the bounce.
But they don't say what this matter IS. Is it electromagnetic radiation? Is it neutrinos? Or what?
Whatever it is, it introduces PERTURBATIONS in the Bunch-Davies vacuum that inflation scenarios postulate existed before inflation.
These MATTER quantum fields are what they label Qˆ, Tˆ. And their Hilbert space is what they call H1. These fields live on a quantum geometry, and its Hilbert space is what they call Ho. So the combination is the tensor product of the two Hilbert spaces. That is where the states of the combined system live.

==quote pages 39 and 40 of http://arxiv.org/abs/1211.1354 ==
Having constructed the dynamics of gauge invariant variables on the truncated phase space, we then used LQG techniques to construct quantum kinematics: the Hilbert space Ho of states of background quantum geometry, the Hilbert space H1 of gauge invariant quantum fields Qˆ, Tˆ representing perturbations and physically interesting operators on both these Hilbert spaces. The imposition of the quantum constraint on the homogeneous sector leads one to interpret the background scalar field φ as a relational or emergent time variable with respect to which physical degrees of freedom evolve. Furthermore, the background geometry is now represented by a wave function Ψo which encodes the probability amplitude for various FLRW geometries to occur. The physically interesting wave functions Ψo are sharply peaked, but the peak follows a bouncing trajectory, not a classical FLRW solution that originates at the big bang. In addition, Ψo has fluctuations about this bouncing trajectory. Quantum fields Qˆ, Tˆ, representing inhomogeneous scalar and tensor perturbations, propagate on this quantum geometry and are therefore sensitive not only to the major de- parture from the classical FLRW solutions in the Planck regime, but also to the quantum fluctuations around the bouncing trajectory, encoded in Ψo. Therefore at first the problem appears to be very complicated. However, a key simplification made it tractable: Within the test field approximation inherent to the truncation strategy, the propagation of Qˆ, Tˆ on the quantum geometry Ψo is completely equivalent to that of their propagation on a specific, quantum corrected FLRW metric g ̃ab. Although h does appear in its coefficients, this ‘dressed, effective metric’ g ̃ab is smooth and allows us to translate the evolution of Qˆ, Tˆ with respect to the relational time to that in terms of the conformal (or proper) time of g ̃ab.

Furthermore, away from the Planck regime, g ̃ab satisfies Einstein’s equations to an excellent approximation. In this sense, the standard quantum field theory of Qˆ, Tˆ emerges from the more fundamental description of these fields evolving on the quantum geometry Ψo with respect to the relational time φ. This exact relation between quantum fields Qˆ, Tˆ on the quantum geometry Ψo and those on the dressed, effective geometry of g ̃ab enabled us to carry over adiabatic regularization techniques from quantum field theory in curved space-times to those on quantum geometries Ψo. Together, all this structure provides us with a well-defined quantum theory of the truncated phase space we began with.
==endquote==


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