# Eading in many books that the unification of quantum mechanics

1. Mar 18, 2006

### Milind_shyani

hi
i have been reading in many books that the unification of quantum mechanics and general relativty is not possible. but why??? it is clear that we cannot use a certain theory for certain phenomen in the universe and other theory for the rest phenomenas of the world.
at singularity we have to consider both quantum mechanics and relativity.but we are no ablr to unify this two theory.but why? why is it so that we cannot prive both the theories together at the same time???

2. Mar 18, 2006

### masudr

Well, how about you say how you think it could be unified? I'm sure most people here will be able to give reasons why any proposed method will have some difficulties.

3. Mar 18, 2006

### DrChinese

Many physicists are working on finding a solution to the problem of unification. It is very difficult, as evidenced by the fact that no one has succeeded so far. Some believe that there is no solution, but no one knows for certain at this time.

4. Mar 18, 2006

Staff Emeritus
When quantum mechanics was extended to quantum field theory, as it had to be to account for measurements, it was discovered that certain calculations led to infinite answers. This was a catastrophe, but a solution was found. What is called the renormalization program can cancel a finite number of these infinities and produce good finite calulations. A theory that only requires a finite number of such cancellations is called "renormalizable", and all the present quantum theories, including the Standard Model of particle phyics, are renormalizable. But if a theory has an infinite number of infinities, the program won't work, and such a theory is unrenormalizable.

General relativity, because its physics has to be invariant under arbitrary diffeomorphisms, and a diffeomorphism has infinitely many degrees of freedom, is unrenormalizable. So it can't be quantized, at least not in the naive straight forward way that suggests itself. This is the core of the unification problem, and the people working to unify the two branches of physics are mostly trying to find a way around it.

5. Mar 18, 2006

### scott1

6. Mar 18, 2006

Staff Emeritus
Renormalization of QFT is discussed at length in all textbooks on Quantum Field Theory, such as Peskin and Schroeder, An Introduction to Quantum Field Theory, Ryder, Quantum Field Theory, and Ramond Field Theory, A Modern primer.

I don't know if it's still available, but John Collins wrote an excellent book called Renormalization in the Cambridge Monographs on Mathematical Physics series.

Most of these also mention the problem with renormalizing GR.

Last edited: Mar 18, 2006
7. Mar 26, 2006

### eljose

-The problems are the "infinities" they appear in QFT perturbative approach, then my doubt is why can not they be cancelled?..suppose that to calculate the mass of a particle we have the series:

$$a_{0}+a_{1}g+a_{2}g^{2}+a_{3}g^{3}+..........$$

the series is infinite but if we find the "optimum" number n given as the minimum of the expression $$|a(n+1)/a(n)|$$ and lets,s say that n=10 for a special case then we can take the sum with only 10 a(n) in this case we would have:

$$a_{0}+a_{1}g+a_{2}g^{2}+............+a_{9}g^{9}$$

then all the constants are infinite but we only have to deal with 10 infinite constants, when you calculate a series you don,t need to sum infinite terms to give a "good" approximation with g the coupling constant

Of course another "solution" to the problem is Path-integral formulation...there exist a derivation of a formula due to Bernoulli in wich you can express the Path-(functional) integral in the form:

$$\int{D[x]e^{iS[x]/\hbar]}=\sum_{n=0}^{\infty}(-1)^{n}(\hbar/i)^{n}\delta^{n}e^{iJ[x]/\hbar}$$

where S is the classical action of the system and J is the action of a particle in wich the potential has an "extra" term of x, where x=x(t) coordinate of the particle.. here the "delta" means the n-th functional derivative operator.

Last edited: Mar 26, 2006
8. Mar 27, 2006

### hellfire

I will give an answer from a different point of view, may be a more simple one: General relativity and quantum mechanics cannot be unified because the first is a classical theory that does not incorporate the principles of superposition and uncertainty, that are basic postulates in quantum mechanics. This leads to a logical inconsistency.

Consider a (quantum) particle with it’s wavefunction, describing a state of superposition on different positions. According to general relativity this particle should have an influence on the curvature of spacetime. It makes not much sense to assign a single definite value of curvature of spacetime for one of the possible positions of the particle, since it’s position is a probability distribution.

On the other hand, it makes also no sense to assign a sum of curvatures of spacetime to the distribution of positions of the particle, because the wavefunction collapse of the particle’s position is instantaneous according to the postulates of quantum mechanics and this would lead to superluminal propagation of the curvature.

What I have mentioned here is described in detail in one of the last chapters of Wald’s book “General Relativity”. Moreover, there is an interesting experiment by Page and Geilker that shows this inconsistency and suggest that spacetime must be quantized in order to have a consistent unification between both.

So you can try to make a quantum theory out of general relativity. Then, depending on the approach you choose, you will run into the problems that the other posters have mentioned.

Last edited: Mar 27, 2006
9. Mar 27, 2006

### vanesch

Staff Emeritus
It was my understanding that one of the hardest problems in the unification of GR and QM is "the problem of time". The axiomatic structure of quantum theory needs a "time axis" ; it doesn't necessarily need a "space axis", but there is a fundamental separation between "time" and "state of the system". The state of the system is described by a ray in Hilbert space, which is supposed to correspond to an "instantaneous" picture of the system, while time is fundamentally a parameter over which the unitary evolution is running.
Whatever are the degrees of freedom in the hilbert space, they can, or cannot, be "space-related" - that's not the difficult part. For instance, we could have some "spin" degrees of freedom which have nothing to do with space ; or we could have particle positions in an Euclidean space (in which case the degrees of freedom are those particle positions Xi,Yi, Zi - promoted to operators over Hilbert space) ; or we could have fields parametrized over 3 (or more) space coordinates x,y,z), this time, the FIELDS become the operators phi(x,y,z), parametrized in (real) x, y and z ; or whatever other exotic set of degrees of freedom you want to consider.
But time is set apart.

This is a priori a very "Newtonian" viewpoint: an "instantaneous" state, and a time evolution. In a Minkowski space, we can still save the day: we can have the time parameter correspond to the parametrisation of space-like planes, and somehow put constraints upon the different states and unitary evolutions, so that there are possible links between DIFFERENT ways of slicing up Minkowski space, so that they are in agreement (this is in fact requiring Lorentz-invariance of this unitary evolution). We can even go further: we can even save the day in a GIVEN CURVED SPACETIME. As long as the spacetime is GIVEN, we can do quantum theory over it, and require that the unitary evolutions, as seen by different "slicings" (and hence different choices of time coordinates) are in agreement.
However, there is a fundamental hick when one wants to introduce the DYNAMICS of spacetime (as is done in general relativity) ; because you bite your tail: in order to set up the quantum theory, you already NEED your spacetime to have your "time parameter" and your "degrees of freedom", while this time parameter is supposed to COME OUT of the dynamics, but it can't come out as long as you don't HAVE your time axis defined (over which the unitary dynamics is to be parametrized).
So the fundamental difficulty comes from the fact that this time parameter enters directly in the fundamental axioms of quantum theory, and is set up totally different from the other degrees of freedom (time = real parameter, other degrees of freedom = structure of Hilbert space). Some people close their eyes and think of England, and START with a given (flat?) spacetime, and define a tensor field over it that becomes ultimately the gravitational curvature ; these approaches are called "background dependent" and are against the spirit of GR ; superstrings suffers from it as far as I understand.

Last edited: Mar 27, 2006
10. Mar 27, 2006

### Ratzinger

11. Apr 3, 2006

### eljose

-You could use "Regge Calculus" and its definition of discrete Lagrangian for Einstein equations..then your "Path integral" is multi-dimensional (but finite)...by the way...could the Montecarlo integration method work for infinite dimensionality?

12. Oct 4, 2006

### djtchong

.....STRING THEORY TO THE RESCUE!!! ~ ~ ~ ~ ~
lol

13. Oct 4, 2006

Staff Emeritus

Yeah, right!