Category Archives: General Mathematics

Can one disprove special relativity with high school mathematics?

Is it possible using mathematics that is not much beyond high school mathematics to prove that special relativity is wrong? And what does that even mean?

The mathematics of special relativity
It is more-or-less true that Einstein’s original works on special relativity do not really use any highbrow mathematics. In a standard undergraduate introduction to the subject no more than linear algebra is really used: vector spaces, matrices and quadratic forms.

So, as linear algebra is well-founded, one is not going to find some internal inconsistencies in special relativity.

Moreover, today we understand special relativity to be based on the geometry of Minkowski space-time. Basically, this is Euclidean with an awkward minus sign in the metric. Thus, special relativity, from a geometric perspective, is as well-founded as any thing in differential geometry.

So one is not going to mathematically prove that special relativity is wrong in any mathematical sense.

On to physics…
However, the theory of special relativity is falsifiable in the sense of Popper. That is, taking into account the domain of validity (ie., just the situations you expect the theory to work), experimental accuracy, statistical errors etc. one can compare the theoretical predictions with what is measured in experiments. If the predictions match the theory well, up to some pre-described level, then the theory is said to be ‘good’. Otherwise the theory is ‘bad’ and not considered to be a viable description of nature.

In this sense, using not much more that linear algebra one could in principle calculate something within special relativity that does not agree well with nature (being careful with the domain of validity etc). Thus, one can in principle show that special relativity is not a ‘good’ theory by finding some mismatch between the theory and observations. This must be the case if we want to consider special relativity as a scientific theory.

Is special relativity ‘good’ or ‘bad’?
Today we have no evidence, direct or indirect, to suggest that special relativity is not a viable description of nature (as ever taking into account the domain of validity). For example, the standard model of particle physics has at its heart special relativity. So far we have had great agreement with theory and experiment, the electromagnetic sector is extremely well tested. This tells us that special relativity is ‘good’.

Even the more strange predictions like time dilation are realised. For example the difference in the life-time of muons as measured at rest and at high speed via cosmic rays agrees very well with the predictions of special relativity.

Including gravity into the mix produces general relativity. However, we know that on small enough scales general relativity reduces to special relativity. Any evidence that general relativity is a ‘good’ theory also indirectly tells us that special relativity is ‘good’. Apart from all the other tests, I offer the discovery of gravitational waves as evidence that general relativity is ‘good’ and thus special relativity is also ‘good’.

The clause
The important thing to remember is that the domain of validity is vital in deciding if a theory is ‘good’ or ‘bad’. We know that physics depends on the scales at which you observe, so we in no way would expect special relativity be a viable description across all scales. For example, when gravity comes into play we have to consider general relativity.

On the very smallest length scales, outside of what we can probe, we expect the nature of space-time to be modified to take into account quantum mechanics. Thus, at these smallest length scales we would not expect the description of space-time using special relativity to be a very accurate one. So, no one is claiming that special relativity, nor general relativity is the final say on the structure of space and time. All we are claiming is that we do have ‘good’ theories by the widely accepted definition.

Are all claims that relativity is wrong bogus?
Well, one would have to examine all claims carefully to answer that…

However, in my experience most objections to special relativity are based on either philosophical grounds or misinterpreting the calculations. Neither of these are enough to claim that Einstein was completely wrong in regards to relativity.

A first look at Lie theory

A friend of mine made a request…

Any chance you could make an expository post on Lie Theory for those of us who only known some abstract algebra and calculus? The topic seems very inaccessible otherwise, but I hear Lie Groups and Lie Algebras mentioned regularly.

As your friendly neighbourhood mathematician I will try to oblige.

Disclaimer What I do is give an informal overview and not worry too much about details and proper proofs. Proofs you can find in textbooks. Rather I want to present the ideas and sketch some constructions.

I will build this account up over the period of a few weeks.

Rough Plan
The things I would like to cover are the following.

  1. Abstract Lie algebras
  2. Lie groups
  3. The Lie algebra of a Lie group
  4. Lie’s theorems
  5. Some odds and ends (Maybe a few words about Lie groupoids etc)

There maybe some changes here as the work develops.

I will also include some simple exercises for those that are interested. I will post solutions at the end.

Part 0: Introduction
Anybody who reads anything about modern physics will encounter the terms ‘Lie group’ and ‘Lie algebra’. Lie theory is all about the relation between these two structures.

A Lie group is a group that also has a smooth manifold structure, importantly the group operations are compatible with this smooth structure. Groups represent transformations and symmetries of mathematical objects. Lie groups are the mathematical framework for studying continuous symmetries of mathematical objects. Thus, Lie groups are fundamental in geometry and theoretical physics.

Now, every Lie group has associated with it a Lie algebra, whose vector space structure is the tangent space of the Lie group at the identity element. The Lie algebra describes the local structure of the group. Informally one can think of the Lie algebra as describing the elements of the Lie group that are ‘very close to the identity element’.

The theory of Lie groups and Lie algebras was initiated by Sophus Lie, and hence the nomenclature. Lie’s motivation was to extend Galois theory, which proved useful in the study of algebraic equations, to cope with continuous symmetries of differential equations. Lie laid down much of the basic theory of continuous symmetry groups.

The plan is with these notes is to sketch the relation between Lie groups and Lie algebras. I will stick to the finite dimensional case for this first look.

Part I: Abstract Lie algebras
Let us start with a completely algebraic set-up. Informally, a Lie algebra is a vector space with a non-associative product, known as a ‘bracket’ that satisfies some nice properties. We will only consider algebras over the reals or complex here, though everything will generalise to more arbitrary fields (with some minor modifications if necessary).

Definition
A Lie algebra is a vector space \(\mathfrak{g}\) together with a bilinear operation \([\bullet,\bullet]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}\), that satisfies the following conditions

  1. Skewsymmetry
    \([x,y] = -[y,x]\)
  2. Jacobi identity

\([x,[y,z]] + [z,[x,y]] +[y,[z,x]]=0\)

for all \(x,y, z \in \mathfrak{g}\).

Note that Lie algebras are non-associative. Thinking of the bracket as a form of multiplication we see that the Jacobi identity is related to the ‘associator’ which is non-zero in general

\([x,[y,z]] -[[x,y],z]= [x,[y,z]] + [z,[x,y]] = [[z,x],y] \neq 0\).

The Jacobi identity can also be written in ‘Loday form’

\([x,[y,z]] = [[x,y],z] + [y,[x,z]]\),

which means that the operator \(Ad_{x}:= [x, \bullet]\) satisfies the Leibniz rule, the so called adjoint operator is a derivation. Note that this form of the Jacobi identity has this interpretation even if the bracket is not skewsymmetric. In fact such bracket algebras are well studied and are usually called “Loday” or “Leibniz-Loday” algebras.

The dimension of a Lie algebra is defined to be the dimension of the underlying vector space. Elements of a Lie algebra are said to generate that Lie algebra if they form the smallest subalgebra that contains these elements is the Lie algebra itself.

Example Any vector space equipped with a vanishing bracket \([x,y]=0\), is a Lie algebra. We call any Lie algebra with a vanishing bracket an abelian Lie algebra.

Example The (real) vector space of all n×n skew-hermitian matrices together with the standard commutator is Lie algebra. This Lie algebra is denoted \(\mathfrak{u}(n)\).

Example The Heisenberg algebra is the Lie algebra generated by three elements x,y,z and the Lie brackets are defined as
\([x,y] =z\), \([x,z] =0\) and \([y,z] =0\).

Given a set of generators \(\{T_{a}\}\) we can define the Lie algebra in terms of its structure constants. As the Lie bracket of any pair of generators must be a linear combination of the generators we have

\([T_{a}, T_{b}] = C^{c}_{ab}\: T_{c}\),

and so the Lie algebra is determined by the structure constants \(C^{c}_{ab}\).

Exercise How many one dimensional Lie algebras are there up to isomorphisms?

Exercise There are exactly two Lie algebras of dimension two over the real numbers, up to isomorphism. Can you write these down in terms of generators?

Exercise What conditions do the structure constants need to satisfy in order to have a Lie algebra? (Hint: think about the two defining conditions of a Lie algebra)

People study Lie algebras in their own right, but historically they arose from the study of Lie groups. From my own perspective, it is the fact that Lie algebras are ‘infinitesimal Lie groups’ that makes them interesting and useful. In the next section I will move on to groups and in particular Lie groups.

Part II: Lie groups
Before we move on to Lie groups, let us recall the notion of a group. Generically, one thinks of groups as encoding transformations and symmetries of mathematical objects, so they arise all across mathematics.

Definition
A group is a set \(G\) together with a binary operation \(\circ: G \times G \rightarrow G\) that satisfies the following axioms

  1. Associativity
    For every \(a,b,c \in G\) we have \((a\circ b) \circ c = a\circ (b \circ c)\).
  2. Existence of the identity
    There exists an element \(e \in G\) such that \(e\circ a = a \circ e\) for all \(a \in G\).
  3. Existence of inverse elements
    For every \(a \in G\) there exists an element \(b := a^{-1}\) such that \(a\circ b = b \circ a =e\).

It can be shown that the identity element \(e\) is unique. There is only one identity element. Note we have said noting about commutativity. Generally \(a\circ b\) is not the same as \(b\circ a\). Groups for which these two expression are always equal are called abelian groups.

Example The set of integers \(\mathbb{Z}\) together with standard addition form an abelian group. The identity element is zero and the inverse of any element is \(a^{-1} = {-}a \).

Exercise Does the set of real numbers \(\mathbb{R}\) equipped with standard addition form a group? Does the set of real numbers with standard multiplication form a group?

Example A symmetric group a set consists of permutations on the given set; ie. bijective maps from the set to itself. The product is just composition of the permutations as functions. The identity element is just the identity function from the set to itself. The inverse of an element is just the inverse as a function.

Example Probably the simplest non-abelian group is the rotation group \(SO(3)\). This group consists of all rotations about the origin of three-dimensional Euclidean space and the composition is just standard composition of linear maps. Because all linear transformations can be represented by matrices (once a basis has been chosen) the group \(SO(3)\) can be represented by the set of orthogonal 3×3 matrices and standard matrix multiplication. This group is non-abelian as the order of which rotations are composed matters.

Now, Lie groups are both groups and smooth manifolds at the same time. Before we make this statement a bit more precise I should say a few words about manifolds…

For an informal overview of the idea of manifolds you can consult an earlier post I made here. I will assume everyone had read this, or is at least familiar with the basic idea. I will review the minimum needed to define a Lie group.

A manifold is a ‘space’ that is locally similar to \(\mathbb{R}^{n}\) for some n. A smooth manifold is a refinement of that notion to allow us to do calculus. Any manifold can be described by a collection of charts, also known as an atlas.

An atlas on a topological space \(X\) (say) is a collection of pairs \(\{(U_{\alpha},\phi_{\alpha})\} \) called charts, where the \(U_{\alpha}\) are open sets that cover the topological space, such that

\(\phi_{\alpha}: U_{\alpha} \rightarrow \mathbb{R}^{n},\)

is a homomorphism of \(U_{\alpha}\) onto an open subset of \(\mathbb{R}^{n}\). Loosley this means that locally we can ways think about cutting our topological space up into small pieces of the real linear space.

The transition maps are defined as

\(\phi_{\alpha \beta}:= \phi_{\beta} \circ \phi^{-1}_{\alpha}|_{\phi_{\alpha}(U_{\alpha} \cap U_{\beta})}: \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \rightarrow \phi_{\beta}(U_{\alpha} \cap U_{\beta}).\)

Any topological space with an atlas is a topological manifold. Loosley, the transition maps allow you to sew together the local patches by telling you what happens on the overlap of such patches.

We will be interested smooth manifolds, that is we insist that the transition maps be infinitely differentiable in the standard sense. Because we can describe everything locally on a smooth manifold in terms of smooth transition functions and local patches of \(\mathbb{R}\) we can extend all our knowledge of standard multi-variable calculus to smooth manifolds.

In particular we know what a smooth map between two smooth manifolds is. As topological spaces a map between smooth manifolds is a continuous map. To define it as ‘smooth’ we compose the function with a chart on our source and target manifolds and as we know what smoothness means for map from \(\mathbb{R}^{n}\) to say \(\mathbb{R}^{m}\) we can accordingly define smoothness for maps between smooth manifolds.

Exercise Fill in details for the above paragraph.

We can now state what a Lie group is…

Definition A Lie group \(G\) is a smooth manifold that also carries a group structure whose product and inversion operations are smooth maps.

That is both

\(\mu : G \times G \rightarrow G\)
\((x,y) \mapsto \mu(x,y) = x\cdot y\)

and

\(inv : G \rightarrow G\)
\(x \mapsto x^{-1}\)

are smooth maps.

Examples to follow…

On a variant of rhodonea curves

Rhodonea curves or rose curves are plots of a polar equation of the form
\(r = \cos(k \theta)\).

If we specialise to equations with

\(k= \frac{n}{d}\)

for n and d integers (>0), then we have plots of the form below. In the table n runs across and d down

Now, just for fun I considered a slight variant of this given by

\(r = \cos( k \theta) – k\)

The plots are as follows

For another variant I considered

\(r = \cos( k \theta) – k^{-1}\)

I am not sure there is anything mathematically deep here, I just like the images and classify this as some basic mathematical art.

Publishing negative results

Recently the journal New Negatives in Plant Science, was launched with the aim of publishing negative, unexpected or controversial results in the field plant biology this.

This journal is aimed at plant science, but I have always thought that some kind of journal in mathematics that presents results that are ‘close but no cigar’ could be useful; for example one could present results of things that at first look should work, but do not. (Everybody’s note book is full of such things!) However, no-one would want to publish results that are not correct. The only way I can see to turn this around is to develop ‘no-go theorems’.

By ‘no-go theorems’ I mean clear mathematical reason why something the community expected to work does not. Such theorems are usually to be found in theoretical physics, but they can appear in pure mathematics also.

Such concrete statements are of course published in standard journals. Examples that spring to my mind are the Weinberg–Witten theorem, Coleman–Mandula theorem and the no-cloning theorem. Plenty of other examples exist.

Link
Why Science Needs to Publish Negative Results

LMS popular lectures

The London Mathematical Society (LMS) Popular Lectures present exciting topics in mathematics and its applications to a wide audience. Because the LMS is 150 years old this year they are having 4 lectures this year instead of the usual 2.

This years speakers are:

  • Professor Martin Hairer, FRS – University of Warwick
  • Professor Ben Green, FRS – University of Oxford
  • Dr Ruth King – University of St Andrews
  • Dr Hannah Fry – University College London

The lectures will be held in London, Birmingham, Leeds and Glasgow.

The topics seem to be catered to the general populous, I won’t expect the opening line to be “Let E be a quasicoherent sheaf of modules on X…”

For more details follow the link below

Link
LMS popular lectures 2015

du Sautoy asks "can anyone be a maths genius?"

Prof. du Sautoy asks this very question.

How many times have you heard someone say ”I can’t do maths”? Chances are you’ve said it yourself.

du Sautoy talking to the BBC

In all honesty I find myself thinking the above at least twice a day.

Genes or hard work
I am not an expert in how genes play a role in our intelligence, but for sure they do. That said, no-one is born an expert in mathematics and it takes a lot of hard work. Like everything in life, becoming proficient in mathematics to the level you set yourself is about perseverance and the willingness to struggle with things until you have mastered them.

Link
Can anyone be a maths genius? BBC iWonder

Two quotes on the philosophy of mathematics

I gave a talk the other day based on our recent work on graded bundles in the category of Lie groupoids. Anyway, as part of the motivation I drew the audiences attention to two quotes…

“Mathematics is written for mathematicians.” Copernicus

“For the things of this world cannot be made known without a knowledge of mathematics.” Roger Bacon

They show the two different sides of mathematics; mathematics motivated by mathematics and mathematics motivated by applications. I think one should to some extent sit in the middle here, but ultimately it is nice when mathematics has something to to with the real world, even if that connection is somewhat loose.

My real motivation for these two specific quotes was that Copernicus was Polish and Bacon English!

Supersymmetry and mathematics

CERN Prof Beate Heinemann, from the Atlas experiment at CERN had said that they may detect supersymmetric particles as early as this summer. But what if they don’t?

What if nature does not realise supersymmetry? Has my interest in supermathematics been a waste of time?

Superysmmetry

We hope that we’re just now at this threshold that we’re finding another world, like antimatter for instance. We found antimatter in the beginning of the last century. Maybe we’ll find now supersymmetric matter

Prof Beate Heinemann [1]

In nature there are two families of particles. The bosons, like the photon and the fermions, like the electron. Bosons are ‘friendly’ particles and they are quite happy to share the same quantum state. Fermions are the complete opposite, they are more like hermits and just won’t share the same quantum state. In the standard model of particle physics the force carriers are bosons and matter particles are fermions. The example here is the photon which is related to the electromagnetic force. On the other side we have the quarks that make up the neutron & proton and the electron, all these are fermions and together they form atoms.

Supersymmetry is an amazing non-classical symmetry that relates bosons and fermions. That is there are situations for which bosons and fermions can be treated equally. Again note the very different ‘lifestyle’ of these two families. If supersymmetry is realised in nature then every boson will have a fermionic partner and vice versa. In one swoop the known fundamental particles of nature are (at least) doubled! Moreover, the distinction between matter and forces becomes blurred!

A little mathematics
Without details, the theory of bosons requires the so called Canonical Commutation Relation or CCR. Basically it is given by

\([\hat{x},\hat{p}] = \hat{x} \hat{p} – \hat{p} \hat{x} = i \hbar \).

Here x ‘hat’ is interpreted as the position operator and p ‘hat’ the momentum. The right hand side of this equation is a physical constant called Planck’s constant (multiplied by the complex unit, but this is inessential). The above equation really is the basis of all quantum mechanics.

The classical limit is understood as setting the right hand side to zero. Doing so we ‘remove the hat’ and get

\(xp- px =0 \).

Thus, the classical theory of bosons does not require anything beyond (maybe complex) numbers. Importantly, the order of the multiplication does not matter here at all, just think of standard multiplication of real numbers.

The situation for fermions is a little more interesting. Here we have the so called Canonical Anticommutation Relations or CAR,

\(\{\hat{\psi}, \hat{\pi} \} = \hat{\psi} \hat{\pi} + \hat{\pi} \hat{\psi} = i \hbar\).

Again these operators have an interpretation as position and momentum, in a more generalised setting. Note the difference in the sign here, this is vital. Again we can take a classical limit resulting in

\(\psi \pi + \pi \psi =0\).

But hang on. This means that we cannot interpret this classical limit in terms of standard numbers. Well, unless we just set everything to zero. Really we have taken a quasi-classical limit and realise that the description of fermions in this limit require us to consider ‘numbers’ that anticommute; that is ab = -ba. Note this means that aa= -aa =0. Thus we have nilpotent ‘numbers’, that is non-zero ‘numbers’ that square to zero. This is odd indeed.

Supermathematics and supergeometry
In short, supermathematics is all about the algebra, calculus and geometry one can do when including these anticommuting ‘numbers’. The history of such things can be traced back to Grassmann in 1844, pre-dating the applications in physics. Grassmann’s interests were in linear algebra. These odd ‘numbers’ (really the generators of) are usually referred to as Grassmann variables and the algebra they form a Grassmann algebra.

One of my interests is in doing geometry with such odd variables, this is well established and a respectable area of research, if not very well represented. Loosely, think about simple coordinate geometry in high school, but now we include these odd numbers in our description. I will only reference the original paper here [2], noting that many other works evolved from this including some very readable books.

What if no supersymmetry in nature?
This would not mean the end of research into supermathematics and its applications in both physics & mathematics.

From a physics perspective supersymmetry is a powerful symmetry that can vastly simplify many calculations. There is an industry here that works on using supersymmertic results and applying them to the non-supersymmetric case. This I cannot see simply ending if supersymmetry is not realised in nature, it could be viewed as a powerful mathematical trick. In fact, similar tricks are already mainstream in physics in the context of quantising classical gauge theories, like the Yang-Mills theory that describes the strong force. These methods come under the title of BRST-BV (after the guys who first discovered it). Maybe I can say more about this another time.

From a mathematics point of view supergeometry pushes what we know as geometry. It gives us a workable stepping stone into the world of noncommutative geometry, which is a whole collections of works devoted to understanding general (usually associative) algebras as the algebra of functions on ‘generalised spaces’. The motivation here also comes from physics by applying quantum theory to space-time and gravity.

Supergeometry has also shed light on classical constructions. For example, the theory of differential forms can be cast neatly in the framework of supermanifolds. Related to this are Lie algebroids and their generalisations, all of which are neatly described in terms of supergeometry [3].

A very famous result here is Witten’s 1982 proof of the Morse inequalities using supersymmetric quantum mechanics [4]. This result started the interest in applying physics to questions in topology, which is now a very popular topic.

In conclusion
Supermathematics has proved to be a useful concept in mathematics with applications in physics beyond just ‘supersymmetry’. The geometry here pushes our classical understanding, provides insight and answers to questions that would not be so readily available in the purely classical setting. Supergeometry, although initially motivated by supersymmetry goes much further than just supersymmetric theories and this is independent of CERN showing us supersymmetry in nature or not.

References
[1] Jonathan Amos, Collider hopes for a ‘super’ restart, BBC NEWS.

[2] F. A. Berezin and D. A. Leites, Supermanifolds, Soviet Math. Dokl. 6 (1976), 1218-1222.

[3] A Yu Vaintrob, Lie algebroids and homological vector fields, 1997 Russ. Math. Surv. 52 428.

[4] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. Volume 17, Number 4 (1982), 661-692.

Mathematical Physics IOP booklet

The Institute of Physics (IOP) has written a report that examines the relationship between mathematics and physics.

There is no completely agreed upon definition of mathematical physics, which is closer to mathematics that physics in most respects. The IOP report puts it like this;

Mathematical physics is best described as consisting of two parts: physical research that proceeds primarily through mathematical means and areas of mathematics that work to solve the problems posed by physics.

The Journal of Mathematical Physics define mathematical physics as;

…the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.

The IOP booklet looks at a wide range of topics in mathematical physics from quantum mechanics, gravity & black holes, random matrix theory, solitons and topological insulators.

Link
Mathematical Physics: What is it and why do we need it?