##### A physicist’s letter of complaint

“I have become Time, the destroyer of your desire to do physics.”

Bhagavad Gita, approximate translation

Every child knows what time is. It’s literally everywhere. Nothing makes sense without it. No one can even imagine a world without it. Time brings us to life and ends us. We are born into time and exist in time.

Appropriately, time pops up everywhere in our theories about the world.

And yet physicists have a tremendously **hard time saying what time actually is**. Time is the ultimate enigma, the Zen Koan of theoretical physics, the scientist’s nightmare: you would much rather do without it, but nevertheless it’s constantly there in front of you, nagging you, calling attention to itself and its contradictions.

But where do these contradictions stem from?

Physics decompose into several overarching theories, which I would for the purposes of this article classify as general relativity, quantum physics, and thermodynamics, each bringing its own conception of time to the table.

These are infamously hard to bring under one hat: physicists have struggled with a unified conception of relativity, quantum physics and thermodynamics for 100 years.

And then there are internal problems of interpretation in each of these branches of physics. What role do we as scientists and observers play in shaping the form of the theory?

I’ll try to give a brief overview of how these different theories conceptualize time. I won’t provide any solutions. I don’t have any. So instead of giving answers, I will focus on what I think is most annoying about time, which there is no lack of either because time is really annoying…

…nevertheless, we are drawn to it like moths to the flame, and for me, nothing quite gives that kick of puzzlement and awe at the incomprehensibility of the world as thinking about time.

## 1. The Arrow of Time

Time is a game

played beautifully by children.

— Heraklitus

There is very obviously an arrow of time. You don’t get up in the evening, watch Netflix, then go to work, come home, eat breakfast and head to sleep. We don’t slowly develop back to being apes living in the savannah (although in some countries this claim might not hold true anymore).

*The arrow of time defines our lives.*

Nevertheless, wiki calls the arrow of time “one of the unsolved general physics questions”. That is because the laws of physics do not seem to care about it. They are symmetric in time, which is known under the name of *Time-Reversal Symmetr** y*. If you look at the world under a microscope, every little process is reversible. Every interaction of the fundamental particles can happen forwards and backward in time. Energy and momentum plus the relevant quantum numbers are conserved (read here for a much more detailed introduction to symmetries in modern physics), and if everything fundamental is conserved,

*nothing really changes.*The arrow of time only kicks in on larger scales, when many particles are involved and thermodynamics define the rules of the game.

The second law states that entropy always increases in every physical process, and thus every physical process indeed has a preferred direction.

But entropy is a quantity that is almost as bad as time when it comes to grasping it. To quote von Neumann, when he counseled Shannon about naming his new uncertainty function:

*“You should call it entropy, for two reasons,” von Neumann told him. “In the first place your uncertainty function has been used in statistical mechanics under that **name, so it already has a name**. In the second place, and more important, **nobody knows what entropy really is**, so in a debate, you will always have the advantage.”*

Is entropy an information-theoretic quantity that only exists within our theory? Is entropy a byproduct of our biased perspective of the world, of the fact that we conceive of the initial state of the universe during the big bang as extremely unlikely, while it might not be as unlikely in other formulations of the theory (see Penrose’s cyclic universe idea)?

Explaining time by introducing another quantity that no one gets seems like a weak trick, but well, what should we do?

## 2. Time Reversal Symmetry is broken, but only a little

Okay, I have been lying, because time symmetry gets broken on the fundamental level.

*But only a little.*

Because while quantum mechanics is still thought to observe time-reversal symmetry on the level of the Schrödinger equation (leaving out measurements for now), this does not hold in a quantum field theoretical context.

One of the foundational theorems to guarantee a renormalizable field theory is that of **CPT symmetry**, which requires that the combination of **C (charge)**,** P (parity)** and** T (time reversal) **symmetry has to be conserved in order to lead to a Lorentz invariant theory.

The details of this aside, CP violations were observed in some Kaon decays by Cronin and Fitch in 1964, which caused quite the stir in the physics community. It constituted a serious challenge to the concept of T symmetry, as, for CP to be broken and CPT to hold, T symmetry has to be broken as well. Even worse: in more recent times, direct observations of T violations have been made in Kaons.

But the experiments showing this are few, and they are from on a small area of particle physics, and what are somewhat exotic particle decays.

Bottom line: the fact that some Kaons that no one really cares about except for the odd particle physicist violate time symmetry is for me even more unsatisfying than if the laws of physics were perfectly symmetric in time and we had only had to deal with the seeming contradiction between quantum physics and thermodynamics.

## 3. Time in Quantum Physics

The quantum world is expressed in the mathematics of Hilbert space, and the physical world is represented by wave functions living

in that Hilbert space. To connect these functions in Hilbert space with spacetime, they can be projected into a representation in position space. As the usual interpretation by Born assigns the square of the wave- function amplitude at point x to the probability of finding the particle at a point x in space, it might be tempting to draw a one-to-one-connection between a point in Hilbert space and in spacetime, but it’s not as simple as that: Hilbert space is not local, and this non-locality has some measurable consequences when working with entangled states, as observed in a large variety of experiments.

This has been made explicit in the Bell inequalities, which extend to time through the Leggett–Garg inequalities. They imply that time is non-local as in quantum mechanics, which manifests in quantum systems being entangled across time.

This non-locality of time allows for some strange, counterintuitive results (explaining them in full detail would, unfortunately, go beyond the scope of this article).

As Aharonov explains here, we can construct experiments in which we measure a quantum system weakly (a good intro on weak measurements can be found here)before projecting it into an eigenstate with a strong measurement. The** outcome **strong measurement** depends on the observer’s choice of axis**, which can also be made a day or a year after the weak measurement.

After obtaining the result of the strong measurement, we can use it to analyze the data of the weak measurement only to realize that **the outcome fits together perfectly with our choice of axis we made days after the weak measurement (**don’t worry if you don’t fully understand this, the argument is rather technical and involves some of the peculiarities of measuring quantum systems**)**.

It has to be noted that this conserves the light-cone structure of special relativity because we can not determine the axis from the results of the weak measurements alone, which would allow us to predict the future. We can therefore not send useful information backward in time, but nevertheless, the whole thought experiment just shows that the role of time in quantum mechanics is quite strange.

Second of all (this point is also a bit technical) time and energy are connected through the time-energy-uncertainty relation, which is a bit like the famous Heisenberg uncertainty relation, but time, unlike space, is not even a quantum mechanical observable, but simply a parameter of the theory. This relates to the fact that constructing a ** Hermitian Time Operator **would cause the energy operator to be unbounded from below.

This means we cannot really measure time in our quantum theory, because all observables are measured through Hermitian operators.

Again what this precisely implies is debatable. But it sure adds to the confusion.

## 4. The role of time depends on the interpretation of QM

As we’ve already seen from the previous example, Quantum Mechanics is really hard to interpret (check my previous articles for more details), and time is in the thick of it.

**explicitly breaks time-reversal symmetry by collapsing the wave function.**You cannot uncollapse the wave function once it is collapsed.

But why we as observers should have the power to break time-reversal symmetry so explicitly is not clear.

If we want to conserve time-reversal symmetry, we need a new interpretation (such as the many-world theory or the two-vector formalism by Aharonov, which is a quantum mechanical theory symmetrized in time) that conceptualizes quantum measurements in a different way.

But as the question of interpretation is still debated, and the physics community strongly divided around it, we also don’t know what role time ultimately plays.

## 5. Nothing happens simultaneously

“Time is what the clock says.”

Albert Einstein

In relativity, space and time get meshed together. Time is not independent of space and movement in space, leading to such mind bogglers as the twin paradox.

The problem with simultaneousness comes right out of relativity. It is a bit different from the other problems with time because in this case, the theory is crystal clear. It’s just impossible to wrap one’s head around what it means.

In relativity, we can not claim that two things are happening at the same time. Because one can always find a different observer in a different frame of reference for which those two things are not happening at the same time, and in many cases even in the reversed order.

So speaking of simultaneousness is logically inconsistent. We cannot say that something is happening right now in this or that far away corner of the universe. This strongly contradicts our intuitions, but if we haven’t thrown our intuitions out of the window by now, it’s our own fault.

## Bottom line

Time must never be thought of as pre-existing in any sense; it is a manufactured quantity.— Hermann Bondi

Time is omnipresent in physics. But as Bondi states, maybe that’s just because we as scientists can’t do without manufacturing time into our theories. In Kant’s Critique of Pure Reason, time is described as one of the necessary modes of our perspective of the world. It is part of our subjective structure and tied to the root question of epistemology that asks what we can know of the world.

Time is crucial for our intuitive understanding of the universe, but in our theories, it eludes our grasp.

This can, frankly, be really annoying, but a wonderful challenge for us to not give up, to dig deeper, and come closer to an understanding of the strange nature of reality.