All of old.
Nothing else ever.
Ever tried. Ever failed.
No matter.
Try again.
Fail again.
Fail better.
This is a quote from Worstward Ho by Samuel Beckett – a poem as impenetrable and opaque as my post on quantization. There is a version of Beckett’s poem with explanations, so I try again, too!
I stated that the description of a bunch of particles (think: gas in a box) naturally invokes the introduction of a hyperspace having twice as many dimensions as the number of those particles.
But it was obviously not obvious why we need those many dimensions. You have asked:
Why do we need additional dimensions for each particle? Can’t they live in the same space?
Why does the collection of the states of all possible systems occupy a patch in 1026 dimensional phase space?
These are nearly the same questions.
I start from a non-physics example this time, because I believe this might convey the motivation for introducing these dimensions better.
These dimensions are not at all related to hidden, compactified, extra large dimensions you might have read about in popular physics books on string theory and cosmology. They are not tangible dimensions in the sense we could feel them – even if we weren’t like those infamous ants living on the inflating balloon.
In Austria we recently had parliamentary elections. This is the distribution of seats in parliament – equivalent to these numbers:
SPÖ (52)
ÖVP (47)
FPÖ (40)
Grüne (24)
Team Stronach (11)
NEOS (9)
Using grand physics-inspired language I call that ordered collection of numbers: Austria’s Political State Vector.
These are six numbers thus this is a vector in a 6-dimensional Political State Space.
Before the elections websites consolidating and analyzing different polls have been more popular than ever. (There was a website run by physicist now working in finance.)
I can only display two of 6 dimensions in a plane, so the two axes represent any of those 6 dimensions. The final political state is represented by a single point in this space – the tip of an arrow:
After the elections we know the political state vector with certainty – that is: a probability of 1.
Before the elections different polls constituted different possible state vectors – each associated with a probability lower than 1. I indicate probabilities by different hues of red:
Each points represents a different state the system may finally settle in. Since the polls are hopefully meaningful and voters not too irrational points are not scattered randomly in space but rather close to each other.
Now imagine millions of polls – such as citizens’ political opinions tracked every millisecond by directly wiretapping their brains. This would result in millions of points, all close to each other. Not looking too closely, this is a blurred patch or spot – a fairly confined region of space covered with points that seems to merge into a continuous distribution.
Watching the development of this red patch over time lets us speculate on the law underlying its dynamics – deriving a trend from the dynamics of voters’ opinions.
It is like figuring out the dynamics of a moving and transforming piece of jelly.
Back to Physics
Statistical mechanics is similar, just the numbers of dimensions are much bigger.
In order to describe what each molecule of gas in a room does, we need 6 numbers per molecules – 3 for its spatial coordinates, and 3 for its velocity.
Each particle lives in the same real space where particles wiggle and bump into each other. All those additional dimensions only emerge because we want to find a mathematical representation where each potential system state shows up as a single dot – tagged with a certain probability. As in politics!
We stuff all positions and velocities of particles into an enormous state vector – one ordered collection with about 1026 different numbers corresponds to a single dot in hyperspace.
The overall goal in statistical mechanics is to calculate something we are really interested in – such as temperature of a gas. We aim at calculating probabilities for different states!
We don’t want to look to closely: We might want to compare what happens if if we start from a configuration with all molecules concentrated in a corner of the room with another one consisting of molecules everywhere in the room. But we don’t need to know where each molecule is exactly. Joseph Nebus has given an interesting example related his numerical calculation of the behavior of a car’s shock absorbers:
But what’s interesting isn’t the exact solution of the exact problem for a particular set of starting conditions. When your car goes over a bump, you’re interested in what the behavior is: is there a sudden bounce and a slide back to normal? Does the car wobble for a short while? Does it wobble for a long while? What’s the behavior?
You have asked me for giving you the equations. I will try my best and keep the intro paragraphs of this post in mind.
What do we know and what do we want to calculate?
We are interested in how that patch moves and is transformed – that is probability (the hue of red) as a function of the positions and momenta of all particles. This is a function of 1026 variables, usually called a distribution function or a density.
We know anything about the system, that is the forces at play. Knowing forces is equivalent to knowing the total energy of a system as a function of any system configuration – if you know the gravitational force a planet exerts than you know gravitational energy.
You could consider the total energy of a system the infamous formula in science fiction movies that spies copy from the computer in the secret laboratories to their USB sticks: If you know how to calculate the total energy as a function of the positions and momenta of all particles – you literally rule the world for the system under consideration.
Hyper-Planes
If we know this energy ‘world function’ we could attach a number to each point in hyperspace that indicate energy, or we could draw the hyper-planes of constant energies – equivalent of isoclines in a map.
The dimension of the hyperplane is the dimension of the hyperspace minus one, just as the familiar 2D planes floating through 3D space.
If energy changes more rapidly with varying particle positions and momenta hyper-planes get closer to each other:
Incompressible Jelly
We are still in a classical world. The equations of motions of hyper-jelly are another way to restate Newton’s equations of motion. You start with writing down Force = mass x accelerating for each particle (1026 times), rearrange these equations by using those huge state vectors just introduced – and you end up with an equation describing the time evolution of the red patch.
I picked the jelly metaphor deliberately as it turns out that hyper-jelly acts as an incompressible fluid. Jelly cannot be destroyed or created. If you try to squeeze it in between two planes it will just flow faster. This really follows from Newton’s law or the conservation of energy!
It might appear complicated to turn something as (seemingly) comprehensible as Newton’s law into that formalism. But that weird way of watching the time evolution of the red patch makes it actually easier to calculate what really matters!
Anything that changes in the real world – the time evolution of any quantity we can measure – is expressed via the time evolution of hyper-jelly.
The Liouville equation puts this into math.
As Richard Feynman once noted wisely (Physics Lectures, Vol.2, Ch. 25), I could put all fundamental equations into a big matrix of equations which I then call the Unwordliness, further denoted as U. Then I can unify them again as
U = 0
What I do here is not that obscure but I use some pseudo-code to obscure the most intimidating math. I do now appreciate science writers who state We use a mathematical crank that turns X into Y – despite or because they know exactly what they are talking about.
For every point in hyperspace the Liouville equation states:
(Rate of change of some interesting physical property in time) =
(Some mathematical machinery entangling spatial variations in system’s energy and spatial variations in ‘some property’)
Spatial variations in the system’s energy can be translated to the distance of those isoclines – this is exactly what Newton’s translates into! (In addition we apply the chain rule in vector calculus).
The mathematical crank is indicated using most innocent brackets, so the right-hand side reads:
{Energy function, interesting property function}
Quantization finally seems to be deceptively simple – the quantum equivalent looks very similar, with the right-hand side proportional to
[Energy function, interesting property function]
The main difference is in the brackets – square versus curly: We consider phase space so any function and changes thereof is calculated in phase space co-ordinates – positions and momenta of particles. These cannot be measured or calculated in quantum mechanics with certainty at the same time.
In a related way the exact order of operations does matter in quantum physics – whereas the classical counterparts are commutative operations. The square bracket versus the angle bracket is where these non-commutative operations are added – as additional constraints to classical theory.
I think I have reached my – current – personal limits in explaining this, while still not turning this blog into in a vector calculus lecture. Probably this stuff is usually not popularized for a reason.
My next post will focus on quantum fields again – and I try to make each post as self-consistent anyway.
elkemental Force 
LOL! I will start with two admissions: (1) Statistical mechanics and vector calculus were, without doubt, the hardest subjects I took in undergrad. 30+ years later and I still have nightmares thinking about preparing for the tests (2) I read this one three times. You have done an excellent job in tackling a very difficult subject and I particularly appreciate the fact that you have successfully made an understandable analogy. Neither elections nor jelly will ever be the same–at least to me–again.
Thanks a lot, Maurice, for taking the time for reading and commenting!
I had a great teacher in theoretical physics to who I owe a lot – someday I need to dedicate a post to him. He had a special way (that I came to appreciate only much later, when comparing other lecture notes…) of introducing all abstract objects (like tensors and vectors) in their most condensed form… never writing indices unless absolutely necessary. Feynman did not agree – his comments about seemingy beautiful vector notation were as sarcastic as the one about Unwordliness I have quoted.
My late professor has written a series of books on theoretical physics that were released in nearly the same year as Feynman’s Physics Lectures – I have often figured that these two series of books are probably characteristic of an ‘American’ modern way of doing physics versus the classical ‘German’ way? I would be interested in comparing more old books to falsify or verify this…
I don’t suppose you’d mind elaborating a bit on the difference between the two…
I think I have some new ideas for future blog posts now :-) Thanks, Maurice!
I was with you up until the subheading ‘Back to Physics.’ I call what you refer to as an n dimensional space, a hyper-volume and the concept is one I’ve discussed before. Imagine you are trying to describe the morphology of an insect. You measure as many things you can think of …. let’s say you end up with 50 variable measures. Each insect you study occupies a position in 50-dimensioned space, the hypervolume. The statistical technique called Principle Component Analysis compresses these 50 dimensions into just two so you may visualize your population of insects as a cloud within a coordinate system. [The technique allows for the resolution of species, for example.] So … my zoological world does overlap, a bit, with your world of physics. After the discussion of hypervolumes however I lost you. I did think the analogy to a political election was brilliant! I’ll try the physics part again! D
I’m intrigued by this use of multidimensional spaces, and subspaces of them, for zoological studies. I’ve been horribly ignorant of them, though; would you know of a decent introduction somewhere?
Joseph – probably this post by Dave (Pairodox Farm) is a primer?
http://pairodox.wordpress.com/2013/05/08/fourth-dimension-surfing/
Of course!! I should have linked your post perhaps! But I add it now:
http://pairodox.wordpress.com/2013/05/08/fourth-dimension-surfing/
This is the correspondence between the two systems:
The coordinates of the state vector in politics are:
(Votes for party 1, votes for party 2, votes for party 3, votes for party 4,…)
The coordinates in phase space in physics are:
(Position of particle 1, momentum of particle 1, position of particle 2, momentum of particle 2,….)
The physics part is then mainly about time evolution of the ‘jelly’ – probably the confusing thing is that we are familiar with the ‘real space’ those particles live in? So there is motion in real space and in hyperspace? We can imagine and compare 1) particles moving in real space versus 2) the time evolution of the corresponding patch in hyperspace… which again looks like something tangible (a piece of jelly) but this is just a mathematical correspondence.
Or is the problem to see where the ‘distribution’ (uncertainty) stems from? In politics it was simply the variance of different polls. In physics it is different possible starting conditions – positions and momenta of particles thrown into a volume of interest. There is a low probability to find them all in the middle of the room and a vacuum in the corners, to these state vectors will be given lower probabilities than the ones representing an even distribution (this is related to entropy… I avoided the term but probably it would have helped?)
Thanks a lot for the comment, Dave!
Antony Flew’s Dictionary of Philosophy is a very handy book. He defines ontology as, “the branch of metaphysics concerned with the study of existence itself (considered apart from the nature of any existent object)”. Someone else, I don’t remember who, said it was the study of the necessary and sufficient conditions of existence. I’m with Nietzsche in thinking ontology is mostly a problem stemming from grammar. But a nagging doubt remains. The copula is an evolutionary success story. Why is that?
If “we assume that the interesting properties in our macroscopic world are determined by probabilities and averages,” as you said, then we would be making an ontological claim. What is being determined and whether a property is objective are the key things. If a property is situated in the thing in the macroscopic world, then it is an ontological claim (probability is a condition of existence). On the other hand, perhaps “property” is a biased-to-the-objective term that refers to something in our theory of the macroscopic world, and not something in that world itself.
Obviously you have touched on something I find very interesting. I had assumed that probabilities determined the macroscopic world until you started these posts. It’s getting more complicated. Thanks for that.
Thanks – this discussion really adds to the post!
I find it hard – though very fascinating – to translate back and forth between concepts in math and those philosophical concepts. [In the past I often had discussions about that causality means (or can mean) in physics and if has any meaning at all.]
I often feel I should add enough disclaimers – such as: This is relevant for classical (non-quantum) systems that have a large number of what physicists call degrees of freedom – as the gas or the shock absorbers in a car. A ball gliding down a perfect friction-less plane does not really “require” a statistical description.
I agree to your definition of property – especially in that context I brought it up. “Temperature” seems so intuitive, but conceptually it is not. It’s very ambiguous probably: On the one hand you could define temperature as what you measure on thermometers – but this is still not a straight-forward property of the world itself, I guess. What you measure on thermometers happens to coincide with some fancy statistical average created by a mathematical crank – seems even more artificial. But what would count as “something in that world itself”? Don’t we use “properties” all the time to describe the world?
If I recall correctly (…as a rule I never google when responding to comments ;-) – I needed to add this after the discussion on David’s blog) philosophers distinguish between sort of primary obvious properties (not sure if properties is the right term) such as spatial dimension and subjective qualia such as color. But hasn’t that become all obsolete after Einstein showed that not even space and time are really that objective and absolute?
Perhaps I’m way off base and your question was rhetorical. In what way may the copula be viewed as an evolutionary success story? Imagine two groups of ancient hominids, one is aware of the concept of self while the other is not. Folks within one group are aware of their existence while folks belonging to the other group are not. The group which is unaware perhaps resembles more closely an even more the ancient common ancestor of hominids and other, less neurally complex, organisms. In order for the group able to cognate self to be considered evolutionary more successful its members must leave more offspring per unit time than members of the other group. In that sense being self-aware may indeed be an evolutionary advantage. Perhaps being self-aware makes one more careful when on the hunt, or more careful when traveling in dangerous environs? Just my two cents … if it’s entirely nonsense, simply ignore it. D
Thanks, Dave – I leave the reply to Steve and will add just a meta-comment:
David (replying below) once said on his blog that he considers the comments better than his posts – this discussion here demonstrates what he means!
I’ve been very busy lately, basically off-line for days, but here are some late comments:
I suppose the evolutionary success of the copula could mean two things. First, as you said, creatures with a sense of self would be more careful. In addition to that, they would also be more social and co-operative. It’s a small step to supposing others have a mind as well.
The second reference is to the evolution of language. I know pretty much nothing about the subject outside of reading a Steven Pinker book, but I think just about every language has a verb ‘to be’. I think it reflects our hard-wired understanding of how the world works. There are things, and things do things or have things done to them. Grammar seems to follow this scheme. In his comments, David Yerle said that it’s interesting that the math fits so well. The same goes for the copula.
After all my fancy thinking, “The copula is an evolutionary success story. Why is that?” It’s nice to wonder about ontological implications, but I’m satisfied to expect that I will never know anything about it.
So … the dimensions are mathematical, and have no ontological implications?
Thanks for your interesting question, Steve! If I am correct in my assumption about what ontological implications are – then yes. But I am not sure – isn’t ontology also about categorizing and classification and structure in a broader sense? Or do I confuse the philosophical meaning of ontology with a more technical one that is used by documentation experts (I am not really familiar with either.).
For example, I could create a ‘space’ of things like WordPress tags, and associate them with number, e.g. based on the frequency they are used by bloggers, and study the dynamics and time evolution. This is again similar to my politics examples. Does this have ontological implications?
Just to chime in, I’d also say they have no ontological implications since all we’re saying is we need a bunch of numbers to represent certain states. What is significant is that we need that number of numbers (?) and that our particular selection of quantities works really well.
That said, in some way all of physics is just a mathematical representation of something which may or may not have anything to do with it. So in a way the mathematical representation does have implications (at least, the fact that it works so well for that piece of reality) and in a way none do.
Some physicists like to say that all we do is mathematical models to predict experiment outcomes. Whether these mathematical models actually resemble reality is an empty question or, at least, a question that cannot be answered scientifically.
Thanks, David! I am still mulling over the role of probability. This view of the world only makes sense if we assume that the interesting properties in our macroscopic world are determined by probabilities and averages. We design that abstract space in order to calculate probabilities or – saying it in another way – this representation is handy because gross statistical averages are ‘the real stull’. Not sure if this is really an ontological implication though.
I am re-reading Nassim Taleb’s book, so that’s probably why I zoom in on probability now. I enjoy his scathing about the (over-)use of statistical physics methods in finance, and he might probably say that advocates of such methodologies are overconfident in their assumptions on ontological implications.
This is off-topic now, but have you ever read The Black Swan or Antifragile? People usually love or hate Taleb’s writings….
Just so you know, dialogs with your WordPress theme keep indenting further right until they can no longer can be read. That’s why I replied to this with a new comment even though that puts the scrolling out of order.
Guilty as charged – it was me who allowed so many levels. Probably I should reduce them again.
David – an off-topic reply:
I tried to reply on your blog, to your discussion with Steve (on being unoriginal), but my comment doesn’t show up. I added a link in two of three attempts so I am probably tagged as a spammer now although I dropped the link in the third attempt. Maybe I shouldn’t have attempted to submit more than once but I was not sure if it was some other glitch (as I still see the Like button as grayed out).
I’m curious, which post is that?
David’s last post on ‘Phone Zombies’ – I tried to reply at the bottom of the discussion that you started with the comment about on ‘style points for remembering’.