Yes, this is a serious physics post – no. 3 in my series on Quantum Field Theory.
I promised to explain what Quantization is. I will also argue – again – that classical mechanics is unjustly associated with pictures like this:
… although it is more like this:
This shows the timelines in Back to the Future – in case you haven’t recognized it immediately.
What I am trying to say here is – again – is so-called classical theory is as geeky, as weird, and as fascinating as quantum physics.
Experts: In case I get carried away by my metaphors – please see the bottom of this post for technical jargon and what I actually try to do here.
Get a New Perspective: Phase Space
I am using my favorite simple example: A point-shaped mass connected to an massless spring or a pendulum, oscillating forever – not subject to friction.
The speed of the mass is zero when the motion changes from ‘upward’ to ‘downward’. It is maximum when the pendulum reaches the point of minimum height. Anything oscillates: Kinetic energy is transferred to potential energy and back. Position, velocity and acceleration all follow wavy sine or cosine functions.
For purely aesthetic reasons I could also plot the velocity versus position:
From a mathematical perspective this is similar to creating those beautiful Lissajous curves: Connecting a signal representing position to the x input of an oscillosope and the velocity signal to the y input results in a circle or an ellipse:
This picture of the spring’s or pendulum’s motion is called a phase portrait in phase space. Actually we use momentum, that is: velocity times mass, but this is a technicality.
The phase portrait is a way of depicting what a physical system does or can do – in a picture that allows for quick assessment.
Non-Dull Phase Portraits
Real-life oscillating systems do not follow simple cycles. The so-called Van der Pol oscillator is a model system subject to damping. It is also non-linear because the force of friction depends on the position squared and the velocity. Non-linearity is not uncommon; also the friction an airplane or car ‘feels’ in the air is proportional to the velocity squared.
The stronger this non-linear interaction is (the parameter mu in the figure below) the more will the phase portrait deviate from the circular shape:
Searching for this image I have learned from Wikipedia that the Van der Pol oscillator is used as a model in biology – here the physical quantity considered is not a position but the action potential of a neuron (the electrical voltage across the cell’s membrane).
Thus plotting the rate of change of in a quantity we can measure plotted versus the quantity itself makes sense for diverse kinds of systems. This is not limited to natural sciences – you could also determine the phase portrait of an economic system!
Addicts of popular culture memes might have guessed already which phase portrait needs to be depicted in this post:
Reconnecting to Popular Science
Chaos Theory has become popular via the elaborations of Dr. Ian Malcolm (Jeff Goldblum) in the movie Jurassic Park. Chaotic systems exhibit phase portraits that are called Strange Attractors. An attractor is the set of points in phase space a system ‘gravitates’ to if you leave it to itself.
There is no attractor for the simple spring: This system will trace our a specific circle in phase space forever – the larger the bigger the initial push on the spring is.
The most popular strange attractor is probably the Lorentz Attractor. It was initially associated with physical properties characteristic of temperature and the flow of air in the earth’s atmosphere, but it can be re-interpreted as a system modeling chaotic phenomena in lasers.
It might be apocryphal but I have been told that it is not the infamous flap of the butterfly’s wing that gave the related effect its name, but rather the shape of the three-dimensional attractor:
We had Jurassic Park – here comes the jelly!
A single point-particle on a spring can move only along a line – it has a single degree of freedom. You need just a two-dimensional plane to plot its velocity over position.
Allowing for motion in three-dimensional space means we need to add additional dimensions: The motion is fully characterized by the (x,y,z) positions in 3D space plus the 3 components of velocity. Actually, this three-dimensional vector is called velocity – its size is called speed.
Thus we need already 6 dimensions in phase space to describe the motion of an idealized point-shaped particle. Now throw in an additional point-particle: We need 12 numbers to track both particles – hence 12 dimensions in phase space.
Why can’t the two particles simply use the same space? Both particles still live in the same 3D space, they could also inhabit the same 6D phase space. The 12D representation has an advantage though: The whole system is represented by a single dot which make our lives easier if we contemplate different systems at once.
Now consider a system consisting of zillions of individual particles. Consider 1 cubic meter of air containing about 1025 molecules. Viewing these particles in a Newtonian, classical way means to track their individual positions and velocities. In a pre-quantum mechanical deterministic assessment of the world you know the past and the future by calculating these particles’ trajectories from their positions and velocities at a certain point of time.
Of course this is not doable and leads to practical non-determinism due to calculation errors piling up and amplifying. This is a 1025 body problem, much much much more difficult than the three-body problem.
Fortunately we don’t really need all those numbers in detail – useful properties of a gas such as the temperature constitute gross statistical averages of the individual particles’ properties. Thus we want to get a feeling how the phase portrait develops ‘on average’, not looking too meticulously at every dot.
The full-blown phase space of the system of all molecules in a cubic meter of air has about 1026 dimensions – 6 for each of the 1025 particles (Physicists don’t care about a factor of 6 versus a factor of 10). Each state of the system is sort of a snapshot what the system really does at a point of time. It is a vector in 1026 dimensional space – a looooong ordered collection of numbers, but nonetheless conceptually not different from the familiar 3D ‘arrow-vector’.
Since we are interesting in averages and probabilities we don’t watch a single point in phase space. We don’t follow a particular system.
We rather imagine an enormous number of different systems under different conditions.
Considering the gas in the cubic vessel this means: We imagine molecule 1 being at the center and very fast whereas molecule 10 is slow and in the upper right corner, and molecule 666 is in the lower left corner and has medium. Now extend this description to 1025 particles.
But we know something about all of these configurations: There is a maximum x, y and z particles can have – the phase portrait is limited by these maximum dimensions as the circle representing the spring was. The particles have all kinds of speeds in all kinds of directions, but there is a most probably speed related to temperature.
The collection of the states of all possible systems occupy a patch in 1026 dimensional phase space.
This patch gradually peters out at the edges in velocities’ directions.
Now let’s allow the vessel for growing: The patch will become bigger in spatial dimensions as particles can have any position in the larger cube. Since the temperature will decrease due to the expansion the mean velocity will decrease – assuming the cube is insulated.
The time evolution of the system (of these systems, each representing a possible system) is represented by a distribution of this hyper-dimensional patch transforming and morphing. Since we consider so many different states – otherwise probabilities don’t make sense – we don’t see the granular nature due to individual points – it’s like a piece of jelly moving and transforming.
Precisely defined initial configurations of systems configurations have a tendency to get mangled and smeared out. Note again that each point in the jelly is not equivalent to a molecule of gas but it is a point in an abstract configuration space with a huge number of dimensions. We can only make it accessible via projections into our 3D world or a 2D plane.
The analogy to jelly or honey or any fluid is more apt than it may seem
The temporal evolution in this hyperspace is indeed governed by equations that are amazingly similar to those governing an incompressible liquid – such as water. There is continuity and locality: Hyper-Jelly can’t get lost and be created. Any increase in hyper-jelly in a tiny volume of phase space can only be attributed to jelly flowing in to this volume from adjacent little volumes.
In summary: Classical mechanical systems comprising many degrees of freedom – that is: many components that have freedom to move in a different way than other parts of the system – can be best viewed in the multi-dimensional space whose dimensions are (something like) positions and (something like) the related momenta.
Can it get more geeky than that in quantum theory?
I said in the previous post that quantization of fields or waves is like turning down intensity in order to bring out the particle-like rippled nature of that wave. In the same way you could say that you add blurry waviness to idealized point-shaped particles.
Another is to consider the loss in information via Heisenberg’s Uncertainly Principle: You cannot know both the position and the momentum of a particle or a classical wave exactly at the same time. By the way, this is why we picked momenta and not velocities to generate phase space.
You calculate positions and momenta of small little volumes that constitute that flowing and crawling patches of jelly at a point of time from positions and momenta the point of time before. That’s the essence of Newtonian mechanics (and conservation of matter) applied to fluids.
Doing numerical calculation in hydrodynamics you think of jelly as divided into small little flexible cubes – you divide it mentally using a grid, and you apply a mathematical operation that creates the new state of this digitized jelly from the old one.
Since we are still discussing a classical world we do know positions and momenta with certainty. This translates to stating (in math) that it does not matter if you do calculations involving positions first or for momenta.
There are different ways of carrying out steps in these calculations because you could do them one way of the other – they are commutative.
Calculating something in this respect is similar to asking nature for a property or measuring that quantity.
Thus when we apply a quantum viewpoint and quantize a classical system calculating momentum first and position second or doing it the other way around will yield different results.
The quantum way of handling the system of those 1025 particles looks the same as the classical equations at first glance. The difference is in the rules for carrying out calculation involving positions and momenta – so-called conjugate variables.
Thus quantization means you take the classical equations of motion and give the mathematical symbols a new meaning and impose new, restricting rules.
I probably could just have stated that without going off those tangent.
However, any system of interest in the real world is not composed of isolated particles. We live in a world of those enormous phase spaces.
In addition, working with large abstract spaces like this is at the heart of quantum field theory: We start with something spread out in space – a field with infinite degrees in freedom. Considering different state vectors in these quantum systems is considering all possible configurations of this field at every point in space!
I have taken a detour through statistical mechanics: Introducing Liouville equations as equation of continuity in a multi-dimensional phase space. The operations mentioned – related to positions of velocities – are the replacement of time derivatives via Hamilton’s equations. I resisted the temptation to mention the hyper-planes of constant energy. Replacing the Poisson bracket in classical mechanics with the commutator in quantum mechanics turns the Liouville equation into its quantum counterpart, also called Von Neumann equation.
I know that a discussion about the true nature of temperature is opening a can of worms. We should rather describe temperature as the width of a distribution rather than the average, as a beam of molecules all travelling in the same direction at the same speed have a temperature of zero Kelvin – not an option due to zero point energy.
The Lorenz equations have been applied to the electrical fields in lasers by Haken – here is a related paper. I did not go into the difference of the phase portrait of a system showing its time evolution and the attractor which is the system’s final state. I also didn’t stress that was is a three dimensional image of the Lorenz attractor and in this case the ‘velocities’ are not depicted. You could say it is the 3D projection of the 6D phase portrait. I basically wanted to demonstrate – using catchy images, admittedly – that representations in phase space allows for a quick assessment of a system.
I also tried to introduce the notion of a state vector in classical terms, not jumping to bras and kets in the quantum world as if a state vector does not have a classical counterpart.
I have picked an example of a system undergoing a change in temperature (non-stationary – not the example you would start with in statistical thermodynamics) and swept all considerations on ergodicity and related meaningful time evolutions of systems in phase space under the rug.