Can the problem of unification of Quantum Mechanics and General Relativity be converted into a pure Mathematical problem?

Your other question was marked as a duplicate of this one, so even though I don't believe they're the same question I'll answer that one here.

For reference, the new question was:

Mathematically show that Quantum Mechanics and General Relativity are inconsistent

Whilst it can't be expressed purely mathematically (it relies on dimensionality arguments), it can be shown mathematically. The trouble is, the actual maths itself is pretty hard to follow at the best of times. I'll try to explain what mathematically goes wrong when you move GR into QM, but it might not look like there's much maths because you just won't understand if I start throwing around propagation integrals.

Firstly, though, you need to realise that quantum mechanics isn't a physical theory in the same way that general relativity is. QM is just a mathematical framework in which one can formulate a theory. Without any physical grounding, QM is just the abstract mathematics of infinite dimensional complex Hilbert spaces. The mathematical framework used for general relativity is the one of classical fields, which is the same framework used for classical electromagnetism.

The approach that has traditionally worked with moving a theory formulated in the framework of classical fields into the framework of QM is called 'canonical quantisation', and you end up with a theory formulated in terms of quantum fields (which is just a sub-framework of QM). When you do this you find that to calculate any probability amplitude (which are the things that a measurable in QM) you have to do a Taylor expansion.

This is because in quantum mechanics time evolution is given by exponentiation of the Hamiltonian (group theoretically, the Hamiltonian is the generator of time evolution). So for some initial state, $|i\rangle$:

$$ |i\rangle \to |i(t)\rangle = U(t)|i\rangle = e^{-i\int_{0}^{t}H(t')dt'}|i\rangle, $$

The definition of such a thing is its Taylor series:

$$ e^{A} \equiv I + A + \frac{A^2}{2} + \cdots + \frac{A^k}{k!} + \cdots $$

In QFT you need to be able to calculate this so you can work out the probability that your initial state evolves into some final state $|f\rangle$:

$$ \mathcal{P}(|i\rangle \to |f\rangle) = \langle f|U(t)|i \rangle^2 $$

Typically, the Hamiltonian ($H$) looks something like:

$$ H(t) = \int d^3x\ g\phi_1(t)\phi_2(t)\cdots $$

where the $\phi_n(t)$ are different fields all interacting and $g$ is coupling constant, which describes the strength of their interaction. This means each term in the expansion looks like:

$$ U_n \sim g^n\ \left(\int d^4x\ \phi_1(t)\phi_2(t)\cdots\right)^n $$

Now, $U(t)$ as a whole must be dimensionless (because it is an exponentiation). This means each and every term in the expansion must be dimensionless. Therefore if $g$ has energy dimension $[g]$ (we're in natural units so all units can be expressed as powers of energy), the integral in the above expression must have dimensions $-[g]$. In general, we don't know if these integrals are going to converge, so we introduce a cutoff high energy scale, $\Lambda$, above which we don't bother integrating. (This is equivalent to choosing a small distance cutoff $L\propto\frac{1}{\Lambda}$ below which we don't integrate.) Then we can examine the behaviour as $\Lambda\to\infty$. The finite cutoff means that the integral will scale like $\Lambda^{-[g]}$.

This means that $U_n \sim g^n \Lambda^{-n[g]}$, and so the rate of change of $U_n$ with respect to $\Lambda$ is given by:

$$ \frac{dU_n}{d\Lambda} \sim -n[g] g^n \Lambda^{-n[g]-1} $$

This rate of change absolutely cannot be positive for quantum fields to be an effective framework to describe the theory. If it is positive, every single term in the Taylor expansion will diverge and you scattering amplitude just becomes infinite. Since $g$, $n$, and $\Lambda$ are all positive, this means:

$$ [g] \geq 0 $$

Note that everything I've said so far is generally true about the entire framework of quantum fields. Therefore, in any theory described in terms of quantum fields, all the coupling constants must have non-negative energy dimension. Note that [g] being zero is also not brilliant: in this case you're highly dependent on the specifics of the theory to save you.

Now let's look at general relativity. The field in question for GR is the metric tensor for a 4-dimensional Lorentzian manifold, $g_{\mu\nu}$ (nothing to do with $g$ the coupling constant). The 'free' case of GR is special relativity, in which case:

$$ g_{\mu\nu} = \eta_{\mu\nu} $$

where $\eta_{\mu\nu} = \pm\mathrm{diag}(-1,1,1,1)$ is the standard Minkowski metric for a flat spacetime. Let's only consider small perturbations, $\delta_{\mu\nu}$ around the free case:

$$ g_{\mu\nu} = \eta_{\mu\nu} + \delta_{\mu\nu} $$

The action (time integral of the Lagrangian) for general relativity is given by the Einstein-Hilbert action:

$$ S = \frac{1}{2G}\int d^4x\ \sqrt{-g}R $$

where $G$ is Newton's gravitational constant, $g$ is the determinant of $g_{\mu\nu}$, and $R$ is the Ricci scalar. Expanding this out you get:

$$ S = \frac{1}{2G}\int d^4x\ \left((\partial\delta)^2 + (\partial\delta)^2\delta + \cdots\right) $$

For this to fit in with the existing framework for QFT we need to rescale $\delta$, so we say:

$$ \delta \to \delta' = \frac{1}{\sqrt{G}}\delta $$

which gives:

$$ S = \frac{1}{2}\int d^4x\ \left((\partial\delta')^2 + \sqrt{G}(\partial\delta')^2\delta' + \cdots\right) $$

Then our Hamiltonian for quantum mechanics is given by:

$$ H = \int d^3x\ \left(\sqrt{G}(\partial\delta')^2\delta' + \cdots\right) $$

Which means our coupling constants are successive powers of $G^{1/2}$, which has energy units of $-1$. Oh dear, all of our coupling constants have negative energy dimension! This is precisely what we wanted to avoid, as it means that the theory is more dependant on higher energies than lower ones; hence, we can't simply ignore high energies. $G \sim [\mathrm{energy}]^{-2}$ because the gravitational potential, $V$, is dimensionless and depends on mass:

$$ V = -\frac{GM}{r} $$

So ultimately general relativity doesn't work as a quantum field theory because its strength scales up with mass and energy, which makes it impossible to apply a high energy cutoff.

That's why quantum fields aren't a good mathematical framework for general relativity. This is telling us that we need a different way to move GR into the framework of quantum mechanics as this naive direct approach yields the wrong theory. Or, maybe we need to develop a new framework that we can move all the theories currently described by QM into along with GR? In any case, such a framework must have a finite 'smallest possible length scale' that provides a natural cutoff for our integral (hello string theory).

A search for a theoretical model of any physics discipline starts by looking at the data/experiments and trying to find a mathematical description that is predictive of new data/experiments. When this is successful the problem is solved because the goal was achieved.

Example: The successful unification of electricity and magnetism, laws and all, by Maxwell's equations.

From the broad mathematical field of second order differential equations , a choice was made, that described data and is accurately predictive:

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio technologies such as power generation, electric motors, wireless communication, cameras, televisions, computers etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents and changes of each other. One important consequence of the equations is the demonstration of how fluctuating electric and magnetic fields can propagate at the speed of light

You ask:

Can we make the problem seem as if it has nothing to do with the real-world and describe it only as a problem of Mathematics?

If there is no description of boundary constraints ( imposed by data) there is no self-contained reason to pick the specific subset of differential equations and their solutions which will describe the physical situation.

The laws and postulates are imposed so as the pick the appropriate subset .

Can the problem of unification of Quantum Mechanics and General Relativity be converted into a pure Mathematical problem?

In a sense it is, with string theories, a lot of mathematics was developed afaik, and it is still at the mathematical stage because the laws or postulates that would pick the correct mathematical subset of the huge number of mathematical string theories so as to have predictive solutions has not been discovered yet.

So I guess my reply is no, because it would make no sense without the postulates/laws.

I mean to say, can you make someone understand the problem who only knows Mathematics but no Physics ? I mean, describing the problem only using the terms of Mathematics and not using a single word related to Physics.


It makes no sense to even try to do that.

Put it this way : I can write down a series of equations which are perfectly sensible mathematically, but have no way of linking them with the real world.

A mathematical system of equations and relations without a connection to the real world is just not a physics model, it's just mathematics - an abstraction.

Let's take a simple case :

$$F = ma$$

It's a perfectly good equation, but if you have no physical meaning for $F$, $m$ and $a$ it's nothing more than a piece of mathematics.

This problems becomes far worse when you deal with GR and QFT because those physical models don't even connect to our common sense easily (it's very hard work just staying grounded in these worlds).

Can we make the problem seem as if it has nothing to do with the real-world and describe it only as a problem of Mathematics?

That's not physics.

The whole point of using mathematics to study the world is to be able to model the behavior of the real world. If you remove the real world, what do you have : just abstract mathematics.

I have to say that I was "brought up" to regard experimental physics as the cornerstone of physics. The goal is, as I was taught, always to find a theory that matches the real world to withing required experimental accuracy, not to construct complicated theories for the mathematical fun of it.

Without experimental evidence leading the way and pointing to flaws in existing theories, we would have no new theories. You cannot divorce the two - it's hand in glove.

And the process of experiment and theory is incremental (or evolutionary, if you prefer). They help each other.

So returning to your question :

Can the problem of unification of Quantum Mechanics and General Relativity be converted into a pure Mathematical problem ?

You cannot divorce physics (even theoretical physics) from the real world, as you have described it.

So a the problem of unification of QM and GR is not doable without reference to the real world.

This problem is much harder for the very simple reason that the process of building such a theory is on the leading edge of out knowledge, both theoretical and experimental. Developments in both will be required to develop a robust theory incorporating concepts from both QM and GR which matches the real world.