Mathematical Abstractions

Clearly, from the quotes right, there is a spread of opinion about the role of mathematics in physics. From a pragmatic perspective, it is obvious that mathematics has played, and will continue to play, a vital role in understanding the wider universe that extends beyond the intuitive experience of our senses. However, it might still be argued that some level of caution might not be misplaced at this point.

"For every problem,
there exists a simple and elegant solution,
which is absolutely wrong."

The scope of the mathematics now used to described quantum field theory is simply too broad and too complex for this website to even attempt. As such, the purpose of the following section of discussions can only act as an introduction to some of the mathematical abstractions, semantics and field concepts that have now come to dominate much of theoretical physics. Whether this level of mathematical abstraction is necessary might also be questioned along the way; although such questioning might well be a reflection of my own ignorance regarding the benefits associated with the modern formulation of mathematics.

The word 'physics' has its origins in the Greek language. In this historical context, the idea of science was closely associated with philosophy, where physics was essentially encompassed within an overarching philosophy regarding the ‘workings of nature’. Although physics was to evolve away from its early philosophical roots due to developments in mathematical formalism; the relationship between physics and mathematics has always remain a matter of philosophical debate. In this context, mathematics has long been described as the language of science, as proclaimed by Galileo in his work ‘The Assayer’ in 1623:

“Philosophy is written in this grand book, the universe which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.”

Of course, since Galileo first wrote these word, the language of mathematics has expanded way beyond ‘triangles, circles and other geometric figures’ to include far more highly abstracted concepts. However, the full scope of this debate may have been brought into sharper focus as people attempted to interpret the implications of quantum physics; for it seems that our current understanding of nature may now rest on a mathematical determination of quantum probability, which possibly cannot be quantified in terms of any deeper physical mechanisms.

Why is this an issue and why is it being discussed at this point?

In part, this question challenges both the purpose and ability of science to provide a description of the 'actual' workings of nature, as opposed to a mathematical model that can only be correlated against some final state of observation. For quantum experimentation appears to suggest that no matter how well we prepared two given systems to some common initial states, the results will always differ on the quantum scale, i.e. determinism is no longer reproducible. For example, we might cite the inability to predict exactly where the next photon will hit the screen in the double-slit experiment or the inability to predict exactly when an unstable nuclei will decay due to its half-life. In essence, the mathematics underpinning quantum physics only allows the probability of the outcome to be predicted.

Note: It should not necessarily be assumed that statistical probability only applies to the quantum realm. Statistical mechanics is used in many fields of modern physics subject to large numbers of variables. For example, it is often used to explain the dynamics of large thermodynamic systems, where temperature is used as a statistical measure of a large number of kinetic collisions too numerous to quantify in terms of the more classical laws of physics.

In the case of Schrodinger’s cat, we can only resolve the 50/50 probability as to whether the cat is alive or dead, after we open the box, not before, such that some might forward the suggestion that the cat is both half-dead and half-alive. As such, there is the perception that the quantum realm is governed by processes, which defy exact prediction beyond a statistical probability. However, because nobody really understands why quantum systems exhibit such apparently probabilistic behaviour, there is a school of opinion that argues that we have no choice but to based our description of quantum reality on the experimental results predicted within the limits of mathematical probability, even though we might not have a ‘mechanical’ explanation as to why. We might summarise this situation as follows:

“Quantum physics does not and cannot describe what is physically happening in the quantum realm; it simply predicting the mathematical probability of a given outcome. As such, the process is an abstraction vindicated by observable results”.

Naturally enough, not everybody holds to this view, therefore, it is possibly worth stopping for one minute to reflect on some of the inherent difficulties associated with trying to get a broad understanding of modern theoretical physics. For it seems that theoretical physics has now had to resort to a great deal of mathematical abstraction in order to just predict the probability outcome of the most fundamental mechanisms of nature. As such, it might be suggested that theoretical physics appears to have two distinct types of practitioners:

Physicists who practice mathematics
Mathematicians who practice physics

However, for most people, either approach can represent what appears to be an impossibly steep up-hill learning curve in regards to the breadth and depth of the mathematics, which has now permeated into all branches of physics – see links below for some examples already outlined.

Of course, just understanding the mathematical concepts, implied by the list above, does not guarantee any real understanding of the physical phenomena associated with these subjects, unless the physics can also be verified. However, in many ways, mathematics has always allowed theoretical physics to run ahead of the ability of experimental physics to verify, such that it might not be unreasonable to table the following question:

How many people actually understand all the mathematical complexity of theoretical physics, as opposed to an ability to reproduce part of it by rote?

While some might feel that the question is above is too negative in its implications, it might be argued that even Richard Feynman, who helped develop Quantum Electro-Dynamics (QED), had his doubts about the depth and breadth of true understanding in this field of research, based on the following quote:

“What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school and you think I’m am going to explain it to you so that you can understand it? No, you’re not going to be able to understand it. Why, then am I going to bother you with all this? Why are you to sit here all this time, when you won’t be able to understand what I am going to say? It is my task to convince you not to turn away because you don’t understand it. You see, my physics students don’t understand it either. That because I don’t understand it. Nobody does.”

Of course, today, this quote might be challenged because it dates back to the 1970’s and much has been learnt over the last 40 years or so. Equally, it is clear that Feynman was an advocate of teaching physics to as broad an audience as possible, although many might not necessarily understand the mathematical premises on which the physics is predicated. However, it would seem that when first Feynman spoke the words above, he had to be describing what was essentially a theoretical mathematical model of quantum physics, which would only later be subject to some degree of experimental verification.

So what point is this introduction trying to make?

For most people, the ability to understand the mathematics of quantum physics will be limited, which includes myself, as the discussion of the following topics may well prove beyond doubt:

Therefore, as indicated, the main purpose of the discussions listed above is simply to provide an initial introduction, which tries to anchor some of the mathematics in more physical concepts. Of course, any in-depth understanding of quantum field theory will require the study of many copious volumes of mathematical theory written on this subject. Therefore, it is accepted that the following discussions will represent little more than a possible starting point for those both interested, and capable, of pursuing the subject to greater depth.