In classical physics we use probability as a numerical representation of uncertainty. Indeed, not only in classical physics: this is our standard way of understanding the world. Whenever a bookmaker quotes an odd, or we say there is a 50-50 chance of Donald Trump creating a diplomatic row with one of his tweets in any given day, we are using probabilities to parametrise uncertainty. Even those who don't understand the details of the mathematical description of probability will still intuitively think in a way consistent with it.
Probability is convenient because probabilities can be used to directly predict frequency distributions. A probability distribution is proportional to the frequency distribution expected on average over a large number of samples. Frequency distributions are natural and intuitive, because all they are is a count of different types of object. And counting is something which even babies know how to do. The mathematics behind probability is thus the same as the mathematics behind counting. And so, probability comes naturally to us.
However, when we come to quantum physics, things are a little bit different, and probability doesn't work as a representation of uncertainty.
In this post, I briefly introduce the axioms behind probability theory, so that we can see in later posts which of them are violated by quantum physics.
I continue my discussion of the quantum theory of uncertainty. Having outlined the axioms of classical theory of uncertainty (probability), I describe which of them don't apply in the real world, and what they should be replaced by.
One of the big philosophical problems in quantum mechanics is the measurement problem. Measurement plays a key role in QM. The simplest interpretation of QM is that there is a wavefunction which describes the likelihood that observers will measure a particle to be in a particular state. This wavefunction has two different modes of evolution in time. The first is a deterministic and smooth evolution, governed by some differential equation (perhaps the Schroedinger equation or Dirac equation). This happens when nobody is looking at the particle. But when somebody looks at the particle, the wavefunction suddenly and indeterminately jumps into one state or another. There are numerous problems associated with this: Why does the observer play such a key role? How do we define measurement? Under what circumstances does the wavefunction collapse? Why is one part of the evolution deterministic and the other indeterminate? And so on. Clearly, something is either wrong with the theory or with the way we are thinking about the theory.
In this post, I take a look at the measurement problem, and whether identifying quantum mechanical states with Aristotelian potentia can help resolve it.
I discuss what is possibly the most famous equation in physics, and some of its consequences.
I now turn my attention to another plank of the mechanical world view, that the world is deterministic. I discuss what the term means, and what it doesn't mean, and what indeterminism does and doesn't imply.
Continuing my introduction to the principles behind contemporary physics, I turn my attention to the next topic. So far I have stated that, to account for the observed interference effects, a physical theory must parametrise uncertainty using amplitudes rather than probability. From observations that the same circumstances don't always lead to the same effect, I have concluded that fundamental physics must be indeterminate rather than determinate. From numerous observation in modern particle accelerators, I have suggested that the premise of mechanism that the fundamental components of matter are indestructible must be false. From considerations about the nature of change, we see that matter must be able to exist in various states, with (to simplify a bit) in a given basis one of these states existing actually and the others existing potentially, with change being the actualisation of a potential state. There is still causality, but it is of a different type of causality to most of those considered: an efficient causality linking one substance with another, and a final causality (in part) listing the possible effects or decay channels of a particle.
However, we still are left with a large number of possible theories of physics. To narrow them down, I now outline the next major premise needed to construct a workable theory of physics. This is possibly the most important advance of twentieth century theoretical physics: a realisation of the fundamental importance of symmetry in physics.
In previous posts, I have discussed some of the axioms of quantum field theory. Now, I begin to turn to how we can apply those axioms to answer real-world problems.
I continue my introduction to QFT by discussing some of the notation used to represent states (potentia) and creation, annihilation and change (actualisation of a potentia). I will then use this in subsequent posts to start showing how we can compute things.
This is all a bit dull, but it is dull in an exciting way. When building a new Castle, people want to see the turrets, gates and great halls. The foundations don't carry the same interest. But if you don't get the foundations right, you won't get any of the exciting bits either. So I just have to ask you to slog through this in anticipation of what is to come.
I continue with my introduction to quantum field theory. Building on my previous post, I look at how to symbolically manipulate operators representing the creation and annihilation of particles.
Once again, this post will be rather technical. I go through the details to illustrate the process of reasoning by which we go from the axioms to the conclusions, and convince the readers that we are not taking any short-cuts in the reasoning. I will summarise what all this means in a later post.
I continue my introduction to Quantum Electrodynamics by putting in place one of the final and most important ingredients. So far, I have presented my basic axioms, described a notation that can represent states of matter and in particular change from one state to another, and shown how we can in principle use this state/operator notation to perform calculations which can then be compared against experiment. Now we need to discuss how the states evolve in time.
My tool to do this will be symmetry. I will demand that the representation used to describe reality satisfies a number of symmetries. Firstly, translation symmetry (the statement that there is no preferred origin of the universe); secondly Lorentz invariance (the symmetry behind special relativity); thirdly scale invariance (the idea that there is no preferred length scale in the universe); fourthly gauge invariance (that only relative and not absolute differences between the phases of likelihoods have physical significance). I combine the notation developed so far, basic observational data (for example the number of space and time dimensions), and the assumption that the likelihood of matter being in certain states has the same symmetry group as a circle. The result is a theory that has been tested to incredible precision and, baring that it is incomplete because it doesn't describe the other forces of nature, has never been refuted.
I continue my introduction to Quantum Electrodynamics, and show how the mathematics described in the previous posts pays off. I discuss the possible interactions between photons and electrons. There are numerous routes from an initial state to a final state consistent with these interactions. For example, an electron can travel from A to B unmolested, or it can emit and absorb the same photon, or it can emit and absorb two photons, or it can emit a photon, which splits into an electron positron pair, which then annihilate each other into a photon, which is then absorbed back into the initial electron. All we observe is the electron at A and what seems to be the same electron at B. We don't know which of these sequence of events happened during the journey. Therefore, to calculate the likelihood that the electron travels from A to B, we have to calculate the likelihood for each individual route, and add these together.
However, quantum field theory is not a description where anything is possible. There are certain rules determining which interactions are possible and which are impossible. I compare these rules against the basic premises of Aristotelian philosophy. What I find is a great deal of consistency between the fundamental axioms of Aristotle's metaphysics and the physics of quantum fields. This suggests that however we interpret quantum physics, the philosophy behind that interpretation needs to be some variation of Aristotelian metaphysics.
It is claimed that quantum switches allow for indefinitely ordered causality, i.e. that you can't tell in principle whether A is the cause and B the effect, or B the cause and A the effect. This, it is claimed, causes a major problem for Aristotlean metaphysics, which depends on a definite causal order.
I answer that one needs to be careful how one defines causality. The version of causality discussed in these experiments is not the same as Aristotlean efficient causality, and thus the quantum switch says nothing against Aristotle's metaphysics.