I've always thought that the reason we had trouble unifying Quantum mechanics and General relativity is:

  • Quantum mechanics is defined "on" space time and time doesn't vary because of energy. Time is considered a constant (or more like an independent variable.) Whereas GR states that energy bends spacetime and that time is not a constant (or it varies).

  • The second thing is that energy is not Globally conserved. While in Quantum mechanics everything is based on energy conservation. And hence, both are not completely compatible.

  • There might be some mathematical incompatibilities like re-normalization which doesn't work for GR. I don't know why though.

Then how does string theory solve these problems? Isn't the theory just an Idea that string exist and their vibrational state determines the type of particle. Then how does it relate to time being considered as a variable, energy conservation problems and space time curvatures?

How does it try to unite GR with QM?

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    $\begingroup$ Related: physics.stackexchange.com/q/387/2451 $\endgroup$ – Qmechanic Mar 20 '17 at 16:13
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    $\begingroup$ Related/possible duplicate: physics.stackexchange.com/q/1073/50583 $\endgroup$ – ACuriousMind Mar 20 '17 at 16:15
  • $\begingroup$ Neither reference to other PSE Q/A's fully answer the question, nor provide details. Eg, the OPS first question, and really the second leads to: since String Theory is one Lorentzian flat background, how do graviton condensates create a curved background at low energy, and what of at high energy? Or , i.e., how does String Theory deal with the non renormalizability? It's not just the it is not an issue that should not bother us. I realize these are hard questions, but the OP is asking how does it try. $\endgroup$ – Bob Bee Mar 20 '17 at 18:51

What you are referring to is called Background Independence.

Among theoreticians, different attitudes towards background independence dominate. Some consider it extremely important (like the founding fathers of Loop Quantum Gravity), others think of it as merely a peculiar feature of the low-energy theory.

The truth is, ofcourse, that any physical theory has to be judged on the basis of predictions that it makes, not on the basis of which approach you find more appealing. This is exactly why quantum gravity research has gone astray a long time ago: the experimental vacuum forces scientists to speculate.

I will wrap it up with a brief overview of how theories like LQG and string theory treat background independence.

LQG tries to capture the insight of Einstein's GR (whcich is exactly that theories of gravity have to be background-independent). It therefore presents a quantization procedure which doesn't make any reference to any specified background.

String theory is originally formulated as a theory of some physical entity (a string) living on the fixed background. The fluctuations of the string are conjectured to behave like the fluctuations of the background in which the string lives. As an indirect proof of this claim: one of the modes in the spectrum of the strings corresponds precisely to the graviton (a perturbation of the background spacetime); RG-flow equations for the worldsheet conformal invariance turn out to imply Einstein's equations for the background spacetime.

String theory is definitely not manifestly background-independent. But this doesn't mean that it isn't background-independent! The question of whether it is background-independent or not is, to my knowledge, still unsettled.

There have been claims made by respected superstring theorists that superstrings might be a first-quantized perturbative version of some background-independent theory. This could be M-theory (though most of the searches for the fundamental formulation of M-theory were carried out in the background-dependent setting, lol); or this could be formulated on the boundary through AdS/CFT.

To conclude: background independence is a beautiful physical insight of General Relativity, there's no doubt about that. But we accept physical theories based on how well they can predict results of experiments, not by how appealing their fundamental principles seem to us. Both superstrings and LQG have yet to give a single numerical prediction verified by experiments (don't take me the wrong way, they give plenty of mutually contradicting predictions, but none of them are experimentally accessible now or in not-so-far future).

Speculations are allowed to satisfy whatever fundamental principles we want them to, really.

  • $\begingroup$ So, can background independent theories solve quantum gravity? I just read about back ground independence... $\endgroup$ – Chandrahas Mar 22 '17 at 14:27
  • $\begingroup$ @Chandrahas what is "solve quantum gravity" exactly? Most people would say that quantum gravity is a well-defined quantum-mechanical theory, which gives General Relativity in some (classical and/or low-energy) limit. Once we adopt this definition, it becomes clear that both background independent and background dependent theories can in fact be quantum gravity candidates. The only requirement that we have is that background independence has to be restored in the limit, because GR is background independent. That being said, I find background independent approaches more aesthetically pleasing. $\endgroup$ – Prof. Legolasov Mar 22 '17 at 23:51

This is an answer by an experimental particle physicist who saw theoretical models developed to fit the particle data from Regge theory and the Fermi interactions ending finally to the present standard model which unifies all three interactions involving elementary particles.

Thus unification of all four forces becomes the holy grail for a theory of everything (TOE) and that is where quantization of gravity comes in.

Effective quantization of gravity is used in cosmological models. Effective because it is only valid over some range of scales , and not generally.

String theories are a good candidate for particle physics , because they contain in the symmetries of the vibrating strings all the group structure of the standard model, which can thus be embedded in a string theory, with the particles being vibrational levels of the universal string. In a sense the whole standard model is a verification of the string model, except that theorists have not managed to propose one unique string theory model, to be tested with new data in accelerator physics. There have been some phenomenological models with large extra dimensions which have not been seen, though they are sought, at the LHC. A second attraction, for particle physics, of string theories, is that they can also accommodate supersymmetry which seems to be necessary theoretically for the standard model . Andas a bonus the successes of the Regge theory can also be incorporated.

Thus the fact that string theories have a spin two elementary particle which can accommodate the graviton ,i.e. encompass quantization of gravity, makes that an attractive candidate for a TOE once a specific model is built from all the thousand possible string theories.

At the moment the other candidates offered for quantization of gravity cannot embed the standard model of particle physics , and thus are not candidates for a TOE .

How does it try to unite GR with QM?

Well, one would have to study how string theories work, after all. The general statement is that gravity can be accommodated in a string theory.


What string theory aims to do is to unify GR with quantum field theory, not quantum mechanics. Before discussing what string theory does, we need to correct some misconceptions you have.

I’m not sure what you mean by time “varying” vs “being a constant”; I assume you mean that in quantum mechanics time is a parameter and space is an operator, and we don’t have Lorentz invariance. However, that comes up when we compare QM with special relativity too, and led to the development of quantum field theory, which fully incorporates special relativity.

Quantum mechanics solves for the behaviour of a system composed of a certain number of particles by finding the eigenstates of a Hamiltonian operator. Quantum field theory does something completely different. Why? Because quantum mechanics quantises classical mechanics, whereas classical field theory quantises classical field theory. Forget everything you know about quantum theory for the moment and just compare these two equations: $$\ddot{\mathbf{x}}=-\boldsymbol{\nabla}V,\, \partial_\mu F^{\mu\nu}=\mu_0 j^\nu.$$Both are equations of motion. The first can be obtained as an Euler-Lagrange equation of the action $$S=\int dt\left( \frac{1}{2}m\dot{x}^2-V\left( x\right) \right),$$while the second can be obtained as one of several Euler-Lagrange equations of the action $$S=\int d^4x \left(-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}-A_\nu j^\nu\right)$$where we abbreviate $d^4x=dtd^3x,\,d^3x=dtdxdydz$. Just as classical mechanics writes an action in terms of functions of time called coordinates and time derivatives thereof (and possibly also time itself, in which case energy is not conserved), so classical field theory writes an action in terms of functions of spacetime called fields and spacetime derivatives thereof (and possibly also spacetime itself, in which case energy is not conserved if in particular there is an explicit time dependence).

What does all this have to do with the Hamiltonian being ditched? Well, when we turn quantisation back on again we discover we can no longer have a probability amplitude for one particle's location. If you rearrange the time-dependent Schrödinger equation as $\dot{\psi}=i\left( \frac{\hbar}{2m}\nabla^2-\frac{V}{\hbar} \right)\psi$, you can use $\rho=\psi^\ast \psi$ to prove that $\dot{\rho}+\boldsymbol{\nabla}\cdot\mathbf{j}=0$ for probability 3-current $\mathbf{j}=\frac{i\hbar}{2m}\left(\psi\boldsymbol{\nabla}\psi^{\ast}-\psi^{\ast}\boldsymbol{\nabla}\psi\right)$. If this probability interpretation can survive in relativity, we need $\partial_\mu j^\mu = 0$ with $\int d^3 x j^0 = 1$ for some $j^\mu$ expressible in terms of solutions of a relativistic variant of the Schrödinger equation. But in theory, the Schrödinger equation can be interpreted as an equation in some field $\psi$ that has no probabilistic interpretation. Indeed, any relativistic upgrade ends up with solutions for which $\int d^3 x j^0 \le 0$. The ultimate resolution is to see $j^0$ as a difference bewteen particle and antiparticle densities and take the fields in the theory as descriptions of the entire population of such particles and antiparticles in the universe (e.g. the Dirac spinor describes all electrons and positrons). But if particles are now measurable properties of fields in the same way we're used to thinking of momentum as a measurable property of one particle, it is the Lagrangian, not the Hamiltonian, that takes centre stage. The theory doesn't need energy conservation to work, but it gets it anyway in Minkowski space (or, indeed, any spacetime whose metric tensor's determinant is time-independent).

Now let's talk about renormalisation. All current field theories are low-energy variants, obtainable as discussed here, of as yet unknown high-energy theories. As explained here, the proof that general relativity isn't a renormalisable quantum field theory boils down to showing that, whereas any renormalisable QFT has entropy-energy relation $S\propto E^{1-d^{-1}}$ at high energies in a $d$-dimensional spacetime, for GR we get a different power law. Which one? It depends on the spacetime geometry considered. The high-energy spectrum is that of a large black hole, which when plonked into an otherwise large-scale Anti de Sitter spacetime gives $S\propto E^{1-\left( d-1\right)^{-1}}$, so the high-energy conformal field theory recovered is missing a dimension. The AdS/CFT correspondence often discussed in string theory places this CFT on the boundary of spacetime. (The observed acceleration of the universe's expansion indicates de Sitter space is a better model of our universe, but there is also a dS/CFT correspondence that string theory can use.) You may have heard people talk of a "holographic principle"; this is what they're talking about.

The paper I linked to first in the above paragraph ends with a brief clarification of some common misconceptions about exactly what is "wrong" with GR. It'd be better to say there are some unusual features of it, so that we have slightly different unanswered questions hanging over it. Although GR isn't renormalisable, we can make some low-energy predictions for its quantisation, e.g. a calculable $r^{-3}$ correction to Newton's potential. We can also predict that, at or before the Planck energy, new high-energy physics must take over.

String theory attempts to be a high-energy theory whose low-energy spectrum recovers all known physics. Instead of modelling particles as point masses, string theory considers them to have some length, which addresses the renormalisability of gravity. String theory requires a number of extra spacetime dimensions to work, but these are invisible at low energies because they're "compactified", i.e. small in extent. If the size of these new dimensions was the Planck length, new physics would occur at the Planck energy. The energy cutoff for new physics is actually a little lower than the Planck energy, because string theory responds to the hierarchy problem (i.e. gravity being much weaker than the weak interaction) by positing compactified dimensions larger than the Planck length.

String theorists still don't know why several dimensions would compactify, or why having done so they would adopt the specific geometry (called a Calabi-Yau manifold) that they did. But string theory predicts that the topology of this manifold determines the laws of physics (including the set of particle species), while the sizes of the holes in this topology determine the parameters in those laws. This can create about 10^500 possible kinds of physics, forming what's often called the string theory landscape. Pinpointing our physics therein is still an ongoing research topic.

If you'd like to learn more about string theory's approach, see here.

  • $\begingroup$ For some reason my links aen't working, but in order they are: arxiv.org/pdf/1702.00319.pdf arxiv.org/pdf/0709.3555.pdf www.damtp.cam.ac.uk/user/tong/string/string.pdf $\endgroup$ – J.G. Mar 21 '17 at 9:11
  • $\begingroup$ These aren't working either. The pdf just won't open $\endgroup$ – Chandrahas Mar 22 '17 at 14:30
  • $\begingroup$ @Chandrahas Which one? All three open for me (Chrome on a PC). Maybe the problem is specific to some device or a browser. $\endgroup$ – J.G. Mar 22 '17 at 16:14

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