# Is a GUT really necessary?

1. Nov 25, 2006

### Lelan Thara

This question touches on both relativity and quantum theory, but it is general enough that I think this is the place for it.

I am aware that the Holy Grail of modern physics is a "Grand Unified Theory", or "Theory of Everything", that will combine general relativity and quantum theory into a cohesive whole. The fundamental issue, as I understand it, is trying to establish a model that successfully quantizes gravity.

My questions are:

Are general relativity and the Standard Model of quantum mechanics contradictory in some way? Do they make contrary predictions where we must assume that one or the other (or both theories) are false unless we accomplish a GUT?

Is there some fundamental reason why we must believe gravity is quantized? Why couldn't we just accept at face value that gravity differs from the other 3 fundamental forces?

Alternately - is the search for a GUT based on something more ephemeral? Is it really a search for theoretical "elegance" - conceptual "beauty"? Do we look for a GUT simply because it feels right that there should be one?

My thanks in advance to anyone who can help clear this up for me.

2. Nov 25, 2006

### DaveC426913

Yes. In a nutshell, GR requires a smooth continuum to space. QM says space is quantized, and further, that Uncertainty increases as your scale decreases. What this means is that if you try to apply to GR at arbitrarily small distances, you basically get energy levels that are arbitrarily large.

QM is an equation, GR is an equation. If you do a substitution, putting GR into QM, the numbers that pop out are infinities. It means that the smallest iota of empty space can contain an infinitely large amount of energy.

Thus, we know that our understanding is incomplete.

3. Nov 25, 2006

### Lelan Thara

That's a remarkably succinct and clear explanation, DaveC - thanks.

Thus far attempts at a GUT have focused on trying to quantize GR, right? Has anyone tried the other approach - getting QM to work in a continuous spacetime?

To phrase it another way - both GR and QED are very successful, correct? Yet the approach towards a GUT seems to be to try to "fix" GR.
Is there a reason for that which can be easily explained?

4. Nov 25, 2006

### Lelan Thara

Another quick question, Dave, that I hope won't sound facetious.

I have read that the results gotten from the equations of QED also result in infinities that require "renormalization" (division by infinity, as I understand it) to provide useful results.

Are these the same infinities that you are referring to that arise when GR is incorporated into QM?

If it's a different set of infinities - why can't they be renormalized like QED infinities are?

Thanks.

5. Nov 25, 2006

Staff Emeritus
Let me just jump in here quickly to tell you that renormalization is NOT "division by zero". Divisions by zero are precisely what renormalization removes. Basically you redefine the scale of your physics by saying that it doesn't - can't be expected to - describe what happens at really tiny scales, which corresponds to really high energies. So you represent this unknown physics in your integrals by constants. You carry these undefined constants through the mathematical developments as you compute your probabilities, and then at the last step you find (if your theory is a renormalizable one) that you can assume your particles have the known experimentally defined mass and so on, and you can "absorb" the constants into these measured numbers. The infinities in quantizing GR are of the same kind as in other field theories, but you have to use an infinite set of constants to do the renormalization trick, and it's impossible to absorb all of them together. So GR is said to be non-renormalizable.

6. Nov 25, 2006

### topovrs

GR and QM have different, incompatible concepts of time. A QM state must be defined for the whole universe at one time. In GR, except in special situations, it is not possible to invariantly define such global instants. These conflicting concepts of time prevent both GR and QT from being used simultaneously in situations where non-trivial aspects of both are relevant.
There is no such fundamental reason. Many theorists believe that QT is more fundamental, and that gravity must thus be quantized, but it is still quite reasonable to seek a fundamental theory in which gravity / spacetime geometry is distinctly different from the quantum particle fields. Various recent approaches to emergent spacetime take this latter path.

7. Nov 25, 2006

### Hurkyl

Staff Emeritus
That's not a requirement for any quantum theory. To wit, LQFT explicitly defines, for any open set U of Minkowski space-time, a set of states over U.

We have empirical evidence of quantization in gravitational contexts.

Last edited: Nov 25, 2006
8. Nov 26, 2006

### DaveC426913

Lelan didn't say "division by zero". Lelan said "division by infinity".

I think the division term is used loosely in this thread, certainly the math is beyond me. But I get the analogy that, whatever renormalization does, the effect is that infinities magically disappear.

9. Nov 26, 2006

### DaveC426913

Keep in mind that any complete theory of the universe will include an understanding of the 3 forces and gravity. It doesn't necessarily mean they must be the same thing.

Imagine we developed a theory of, say, light. We have a model for all light from red, orange, yellow and green, through to blue light. Our model does not explain violet light, so we have another theory for violet. The two models could be completely different - no problem with that. But if we don't understand why violet is different from the other kinds of light, then you must concede that we don't have a complete model of light. In fact, once we find out how violet is different, that difference becomes part of our theory - our unified theory. See?

So: a complete GUT does not necessarily have to make gravity the same as the other forces, it merely has to include an explanation for how it is different.

Last edited: Nov 26, 2006
10. Nov 26, 2006

### vanesch

Staff Emeritus
Yes, but this has nothing to do with quantization of gravity itself. Although it is quite an experimental feat and I respect the work of my collegue quite a lot, what is quantized is simply the neutron, and gravity enters as an entirely classical potential. In fact, the principle is no surprise and was already established by neutron interferrometry (as for instance explained in Sakurai), where the classical gravitational potential enters into the Schroedinger equation.
What is spectacular with Nesvizhevsky's work (apart from the experimental challenge! I can tell you that about every single aspect of the experiment needed some clever tricks), is:
1) the fact that we now have a bound state (at least in one direction)
2) the length scale on which this operates (tens of micrometers).

You could, for instance, compare this to calculating the spectrum of the hydrogen atom (bound state of a quantized charged particle in a Coulomb potential). Although it indicates that the EM field enters into the Schroedinger equation as a potential, it tells you nothing about about any quantization of the EM field itself.

11. Nov 26, 2006

### vanesch

Staff Emeritus
Both quantum theory and GR have internal difficulties: quantum theory has "renormalization" problems, and GR has singularities, which somehow indicate that they cannot be "the final word".

Apart from that, GR and quantum theory offer totally different "world ontologies".

However, the fundamental difficulty resides IMO in 2 aspects which are related. The first, as pointed out here before, is "the problem of time". Quantum theory has a rather "traditional" structure where there is a "state" and an "evolution". This can be adapted to special relativity, where the "state" and the "evolution" transform according to special rules (representations of the Lorentz transformation). However, it will be difficult in this way to make time a dynamical quantity, which is required by GR. It is only when the "background is given" that one can split things in "a state" and "an evolution".

The other aspect is the fundamental incompatibility between a classical theory and a quantum theory. The quantum theory allows for superpositions of states. So the quantum state where the earth is both at its current place, and twice as far from the sun, exists in principle. There's no way to implement this in GR, because in GR one has to "decide" where it is. Only in a hypothetical quantum version of gravity can you have a smooth connection between the quantum theory allowing for superpositions of energy/mass/momentum states and the gravitational interaction. However, this will then allow for a kind of "superposition" of "times", so which time do we now take as the time in the Schroedinger equation ?

In usual applications of quantum theory, this doesn't play any role, because the gravitational effects of superpositions needed are utterly neglegible (the superposition of the positions of electron positions in a molecule have absolutely no observational consequence gravitationally). But if we apply quantum theory all the way, then it is thinkable to have superpositions of states which are "gravitationally different".

Quantum evolution needs some kind of "classical time" for its unitary evolution, but this classical time is not compatible with GR and a superposition of mass-energy tensors as allowed for in quantum theory.

There are different suggestions to get out of this mess, up to date none really totally successful. But here are the reasons I guess, why people think that the current state of affairs needs a fix.

12. Nov 26, 2006

Staff Emeritus
What Patrick just said is absolutely right, and I think all the various contending schools of Beyond the Standard Model would agree. One other constraint is that bothe the pure quantum thoery of the standard model and the classical general relativity have passed many many tests and been as successful as any theories in history. So people don't really want to give them up or develop some unification that contradicts them at the low energies they've been tested at. This is a big constraint on any proposed GUT.

13. Nov 26, 2006

### Lelan Thara

As DaveC pointed out, I did say, "division by infinity", not division by zero. But that aside, you explanation helps me understand renormalization a bit better, and I accept what you're saying - that GR is non-renormalizable.

The most fundamental thing I am taking away from this discussion is that the search for a GUT is not a search for "elegance" - there are real problems to be solved. That's the major thing I wanted to know.

But please, carry on - this is very interesting!

14. Nov 26, 2006

### turbo

Could this experiment be viewed as a measurement of the fine structure of space-time? In other words, perhaps gravity can be continuous, not quantized, but neutrons have a higher probability to exist at discrete intervals defined by the texture of space.

15. Nov 26, 2006

### bananan

How well do you think string theory as a theory of quantum gravity addresses these fundamental issues, as opposed to lqg?

How does the graviton s-matrix in string theory provide a quantum theory of gravity and what sort of ontology does it address, as opposed to LQG's background independence?

Last edited: Nov 26, 2006
16. Nov 26, 2006

### Lelan Thara

How could it be that this texture would affect neutrons but not texturize gravity, since the mass of the neutrons is contributing to the gravity?

17. Nov 26, 2006

### topovrs

Certainly, the neutrons must contribute to gravity. But the product of Newton's constant and the neutron mass is so small that any contribution to spacetime curvature is unmeasurable. By contrast, the affect of the gravitational potential of the entire earth on a neutron is certainly measurable, and was demonstrated in the cited experiment to be no different than any other classical potential. The observed discrete quantum steps confirm the quantum nature of the neutron, but tell us nothing about the quantum structure of space.

With no empirical data, how a non-local quantum state contributes to gravity remains a subject of theoretical speculation, apart from the clear requirement that general relativity must emerge in the classical limit. Claims that gravity must be quantized, and particular approaches to such quantization, are based on theoretical models whose viability remain to be demonstrated.

18. Nov 27, 2006

### vanesch

Staff Emeritus

Exactly, that's what I wanted to say too

19. Nov 27, 2006

### vanesch

Staff Emeritus
I could say some things, but I don't consider myself qualified enough to do so (although I noticed that talking with assurance about things one is not qualified is a key to success... )

20. Nov 27, 2006

### arivero

Indeed. A metaphor : suppose you define naively a "percentual variation of a function" as f(x+d)/d. Fine, but this quantity diverges when d approaches zero. The solution is to substract a counterterm f(x)/d.

(the example can be sophisticated in ways that even the initial definition seems to have sense. Suppose you want to study a variation of some object starting with a value unity at a given initial time. Then you define $$f(t_i+ \delta t) \over f(t_i) \delta t$$ and for infinitesimal time intervals you must remove the divergence with the counterterm $$-1 / \delta t$$. Of course you get $$f' / f$$ evaluated at $$t_i$$)

Last edited: Nov 27, 2006