Categorical Aside
After a few lectures of basic category theory, people often become quite enthusiastic about discussing their favourite examples of objects and morphisms, as if recognising a category for what it is will be magically useful somehow. After a few more lectures their enthusiasm is usually dampened by the obtuseness of it all, and the realisation that just lumping things into categories doesn't really get one anywhere.
And then, after learning some more tricks, there is a tendency to apply these tricks to the same old examples that came up in the first place. For example, we often discuss the category of (finite dimensional) vector spaces over a field $\mathbb{F}$, where it doesn't really matter what $\mathbb{F}$ is, because the only structure given to the category is the basic properties of a vector space, and its ability to be tensored with other spaces. So we might as well be discussing the category of vector spaces over $\mathbb{F}_{2}$, the field with two elements.
In M Theory, lumping everything into an arbitrary well-known category (or functor category) is analogous to deciding that path integrals for quantum gravity should rely on classical geometry: it amounts to making a ridiculously unacceptable assumption about Nature's way of doing geometry. The category theory itself should provide the geometry. Alternatives tend to be tricky, and require delving into axiom systems, or obscure logic and philosophy, but calculating is eventually meant to be easy!
And then, after learning some more tricks, there is a tendency to apply these tricks to the same old examples that came up in the first place. For example, we often discuss the category of (finite dimensional) vector spaces over a field $\mathbb{F}$, where it doesn't really matter what $\mathbb{F}$ is, because the only structure given to the category is the basic properties of a vector space, and its ability to be tensored with other spaces. So we might as well be discussing the category of vector spaces over $\mathbb{F}_{2}$, the field with two elements.
In M Theory, lumping everything into an arbitrary well-known category (or functor category) is analogous to deciding that path integrals for quantum gravity should rely on classical geometry: it amounts to making a ridiculously unacceptable assumption about Nature's way of doing geometry. The category theory itself should provide the geometry. Alternatives tend to be tricky, and require delving into axiom systems, or obscure logic and philosophy, but calculating is eventually meant to be easy!
posted by Kea | 7:33 PM
2 Comments:
"In M Theory, lumping everything into an arbitrary well-known category (or functor category) is analogous to deciding that path integrals for quantum gravity should rely on classical geometry, ..."
Do you have specific examples of people with such confused thinking in quantum gravity?
One example I say which set me thinking was chapter I.5, ‘Coulomb and Newton: Repulsion and Attraction’, in Professor Zee’s book Quantum Field Theory in a Nutshell (Princeton University Press, 2003), pages 30-6.
Zee starts by (non-)quantizing Coulomb's law (write down a Lagrangian consisting of Maxwell's classical equations with mass terms for photon, put that into a Feynman path integral, evaluate the action and show that this leads to an always-positive potential between two similar charges; hence, similar charges repel). He then moves on to (non-quantized) quantum gravity, writing a 5-component tensor representing the 5 polarizations of the graviton in the Lagrangian (assuming spin-2 gravitons, which should have 1 + 2^2 polarizations), evaluates the path integral for that Lagrangian and finds (as expected) that the potential energy between two lumps of positive energy density is always negative, so masses attract.
This is a fiddle for two reasons. First, nobody has ever seen a spin-2 graviton. They are merely assumed, based precisely on the calculation showing that they should always provide an attractive force between positive energy densities or masses. Hence, it is circular logic to calculate that quantum gravity based on spin-2 gravitons is always attractive. It's because you get that result for a spin-2 vector boson, that mainstream people think gravitons are spin-2.
In addition, the procedure is really just a very slight modification to classical physics, and is not a real quantum field theory. There is no mathematical expression describing individual quanta interactions there; just categories of interactions (each Feynman diagram represents a category of quantum interactions).
In quantum electrodynamics, the path integral relies on classical geometry because Maxwell's classical field equations (differential equations modelling continuously variable fields, not quantized fields) with similarly non-quantized terms for the mass of the field quanta, are stuck into the Lagrangian which in turn goes into the path integral.
Is this problem (classical field geometry being used in path integrals) the actual kind of problem you are referring to?
It you want a true mathematical model of air pressure, you can't say it's a constant 14.7 pounds per square inch, because on the smallest area it's not constant; instead it's quantized into chaotic, randomly timed strikes of individual air molecules having unpredictable speed and direction. The mathematical concept of air pressure just averages out the chaotic air molecules strikes. On large scales, it's a useful statistical approximation.
But that model breaks down on small scales, where the statistical approximation (constant pressure) leads to deterministic predictions, while in fact molecular impacts occur at random in chaotic style and prevent determinism.
Most of the equations in quantum field theory, including the Lagrangian and the path integral into which the Lagrangian is inserted, are completely non-quantized statistical models, valid only approximately for large interaction numbers.
I think that the perturbative expansions of terms you get from path integrals, where each term corresponds to a "Feynman diagram", is really categorization of interactions, not true quantization. Each Feynman diagram represents a category of interactions.
In actual fact, most of the interactions which are normally going on correspond to the very simplest Feynman diagrams, i.e. simple interactions occur many times per second to particles, while more subtle interactions (higher order perturbative corrections) occur less frequently.
Because the biggest contributions to common quantum processes are usually due to the simplest Feynman diagrams, there is no reason to suppose that quantum mechanics is particularly weird. As Feynman points out, the biggest contributions to most path integrals occur from interactions occurring very close to the classical model:
‘Light ... uses a small core of nearby space.’ - R. P. Feynman, QED, Penguin, 1990, page 54.
‘When we look at photons on a large scale ... there are enough paths around the path of minimum time to reinforce each other, and enough other paths to cancel each other out. But when the space through which a photon moves becomes too small ... these rules fail ... The same situation exists with electrons: when seen on a large scale, they travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that there is no main path, no ‘orbit’; there are all sorts of ways the electron could go, each with an amplitude. The phenomenon of interference [from Brownian-motion type impacts of individual gauge bosons] becomes very important, and we have to sum the arrows to predict where an electron is likely to be.’- R. P. Feynman, QED, Penguin, 1990, page 84-5.
Those people who claim that quantum theory is 'weird' have an easy excuse not to investigate the possibility that it is not weird but just very simple, i.e., mainly due to simple interactions (the most basic Feynman diagram).
What's weird is that people ignore the fact that complex Feynman disgrams only in general give rise to very small perturbative corrections. Modern physics has very little weirdness, just a bit of chaotic randomness on small scales due to individual (but usually very simple) quantum interactions.
So many people are drawn to physics for physically false reasons (e.g., believing that it shows that the world is weird or complex), that just stating the factual evidence for underlying simplicity is a "heresy" in itself.
Ambiguous results from experiments like Aspect's experiment, are tken by these people to prove that particles are "magically" entangled, but when you look at the experiment it's actually interpreted using physically flawed mathematics. It's based entirely upon wavefunction collapse stuff, but as Thomas Love has pointed out:
"The quantum collapse occurs when we model the wave moving according to Schroedinger (time-dependent) and then, suddenly at the time of interaction we require it to be in an eigenstate and hence to also be a solution of Schroedinger (time-independent). The collapse of the wave function is due to a discontinuity in the equations used to model the physics, it is not inherent in the physics."
If you get a proper quantized quantum field theory, say a Coulomb potential where actual exchange of individual field quanta between electron and proton occurs randomly in time, then you'll end up with non-deterministic electron orbits which (over long periods) can be statistically modelled by the Schroedinger wave equation. (The time-independent Schroedinger wave equation is easy to derive this way, just by analogy to a classical wave arising from an ensemble of particles such as field quanta interactions.)
In other words, classical models of the atom only need a slight correction (the introduction of an electric field composed of radiation quanta being exchanged between charges).
The production of quantum gravity doesn't need that much change to existing ideas, just corrections and modifications to allow for the actual quantum interactions occurring.
To make the vast amount of graviton interactions in the universe subject to simple mathematical modelling, the near symmetry of the universe in any radial distance from us can be exploited. All you have to do then is to model the distribution of mass and other relevant parameters (like recession velocities) as a function of radial distance, and the maths becomes relatively simple. Gravitation then focusses on asymmetries to the normal radial symmetry, caused by things like large fairly nearby masses: Earth, Sun, etc. The problem is that too much belief exists in the complexity of the world for most people to bother with simple models. The mainstream actually uses a lot of classical physics where it claims to be doing quantum field theory, while at the same time claiming that the universe is beyond simple understanding...
A better way to put my argument against the spin-2 graviton is as follows.
The claimed reason to have a spin-2 graviton is the claim that you need a 5 polarization gauge boson in order to have an always-attractive force between two regions of mass-energy.
This is a false model because you never have two regions of energy: gravitons are not just being exchanged between an apple and the Earth. They are being exchanged with all the other masses in the universe around us as well.
Therefore, the failure of spin-2 gravitons (and a massive chunk of the failure of string theory, too) is the lie that you can analyse quantum gravity by ignoring 99.999... % of the mass involved in exchanging gravitons.
By altering the Feynman path integral formulation to include all the masses in the entire universe which are exchanging gravitons (and not falsely restricting the analysis to two regions of positive energy), the need for a spin-2 graviton with 5 polarization tensor is eliminated. Hence, gravitons must be spin-1 radiation.
Extending this many-charge analysis to electromagnetism (instead of the usual treatment that considers a path integral between just two charges), there now must be two types of electromagnetic gauge boson radiation in order that all charges of like sign can exchange such radiation with other charges of like sign, and in order to incorporate the attraction of unlike charges and repulsion of like charges.
Does this seem any clearer? Maybe when I've published the maths, it will.
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