It is pretty common for us to hear about interactions between mathematics and physics. If one search for such phrase on the web, one will find many articles that delve into such a topic. Perhaps it is not that surprising the close link between physics and mathematics as they co-evolve earlier under the name of natural philosophy (see Evelyn Lamb’s Symmetry article “The Coevolution of Physics and Math“). The term natural philosophy has gone out of fashion due to the divergent paths made by mathematics, physics and philosophy itself (see problem of calls for return to natural philosophy by Arran Gare in his article, “Natural Philosophy and the Sciences: Challenging Science’s Tunnel Vision“). Thus having physics and mathematics emerge as separate domains but yet interacting in some nominal fashion, may not be surprising.
So I find it refreshing to find an article about tension between mathematics and physics instead i.e. Miklos Redei, “On the Tension Between Physics and Mathematics“, J. Gen. Phil. Sci. 51 (2020) 411-425. Redei is a Professor of Philosopher of Science at LSE and has written numerous articles on philosophy and foundations of physics. On my wishlist is his book on “The Principle of the Common Cause“, co-authored with G. Hofer-Szabo & L.E. Szabo. Why are tensions more interesting? They tend to drive new ideas in both fields of physics and mathematics. Redei classified the tensions into two classes:
- Tension of Type I: Unavailability of ready-made mathematical concepts needed for physics;
- Tension of Type II: Availability of mathematical concepts but dismissive of rigour for further development pf physics.
In the former, it is clear that there is a need for new mathematical ideas and the historical examples given are calculus for Newtonian mechanics and spectral theory of self-adjoint operators for quantum mechanics. The other two examples are perhaps the interesting ones as they are yet unsolved, namely, ergodic theory for statistical mechanics and operator-valued tempered distributions for quantum field theory. The case of quantum field theory is not surprising and there are many attempts of putting quantum field theory on rigorous basis by axiomatizing or through constructive quantum field theory. A relative recent review on CQFT can be found in Summers’ “A Perspective on Constructive Quantum Field Theory” (arXiv: 1203.3991). For ergodic theory, this is lesser known to me and Redei mentions the reference of Szasz’s “Boltzmann’s Ergodic Hypothesis, A Conjecture for Centuries” (book is open access).
For Type II Tension, my first impression is that it is not totally unrelated to Type I, with quantum field theory come to mind. For instance, functional integrals are often used in field theories without rigour (see here). Redei mentions the Jaffe and Quinn article “Theoretical Mathematics: Towards a Cultural Synthesis of Mathematics and Theoretical Physics” which sparks debates. See responses here. How tension II can lead to progress is not as clear as tension I but mathematical rigour can always uncover new problems that may lead to new progress. An example that Redei mentioned is von Neumann’s own dismissive behaviour over the Hilbert space formulation of quantum mechanics in favour of an operator algebraic approach. Redei wrote an article on this: Why John von Neumann Did Not Like the Hilbert Space Formalism of Quantum Mechanics (and What He Liked Instead). I’m recalling that Bob Coecke always made reference to this von Neumann’s reaction in favour for a more intuitive categorical/pictorial approach. Another reference that was also mentioned is that of Valente’s “John von Neumann’s Mathematical “Utopia” in Quantum Theory“. So will need to read these papers to understand von Neumann’s standpoint.
be two sets. A mapping 
from 
to 
denoted as 
is a subset of 
such that for every 
, there is a unique 
for which 
i.e.
.
or 
. The element 
is called the source and 
is called a transformation of 
is also called the graph of the function 
defined by
. Then the inverse image 
is
.
does not exist and hence giving the empty set.
. The inverse image of 
is
,
, its inverse image is
.
of 
. Note that 
with
, where 
is said to have 
arguments. One can further generalize this by making each argument coming from different spaces i.e.
,
for some 
. An important map of this nature is the projection map
. The projection 
essentially gives the ith component of the domain.
and 
are maps, then the composition map 
is defined by
.
and 
are maps. Then
.
.
.
i.e.
.
has the property
.
, the bijective map 
is called a transformation or permutation of 
where
.
.
.
are bijections (with codomain and domain), then so is 
with the inverse
.
and 
.
,
.
as follows:
.
which are by definition characteristic functions of (other) subset 
.
are in one-to-one corresponding of all possible maps 
.
be an equivalence relation on a set 
from the set 
can be given by![\varphi(x)=[x]_R](https://s0.wp.com/latex.php?latex=%5Cvarphi%28x%29%3D%5Bx%5D_R&bg=ffffff&fg=000000&s=0&c=20201002)
.
can be introduced via transformation/permutation 
or 
:
and 
, then
are surjective 
is surjective;
be maps. Then
;
;
.
or 
, for which the former is called discrete dynamical systems where evolution rule is usually difference equation, while the latter is called continuous dynamical systems with the evolution rule described by ordinary differential equation. One can consider to discretize the state space itself to give cellular automata with their update rules (see also 
rules). It was 
is a function on the state space 
.
can now be treated as a vector field (as one does in usual (local) coordinate-based differential geometry). Vectors field (despite its local coordinatization whose transformation law is known) encodes geometric information of the state space it lives on. The easiest way to see this is the exemplification of the 
forming the phase space 
. Formulating the mechanical system as 
is the Hamiltonian of the system. It is convenient to rewrite this equation using another algebraic structure known as Poisson bracket which is defined as
.
.
and 
and the Hamilton’s equation as
,
is a covariant antisymmetric tensor known as the symplectic form and
.
and 
. (Note: at times, I will not write out the indices and should be understood contextually.) A stationary solution is the one that obeys 
i.e. doesn’t change with time. Of related interest are fixed points 
such that 
; also called equilibrium points. Points 
are called singular points of the vector field 
; also called stationary orbits.
for which
,
), one has
.
of 
will play in the important role of long-term behaviour of the solutions.
to eigenspace of 
.
;
;
,
. With respect to these spaces, one can actually classify the equilibrium points. The equilibrium point is a sink if 
; a source if 
; a hyperbolic point if 
; and an elliptic point if 
. Note that one has a richer variety of equilibrium points than the one-dimensional case simply because there are more ‘directions’ to consider in higher-dimensional cases (characterised by the eigenvalues of 
(borrowing diagram from Berglund):
(arrows either pointing in or out). Case (b) is a saddle point in which 
. Cases (c) and (d) happen when 
and hence giving rotational (or oscillatory motion in phase space). Cases (e) and (f) are more complicated versions of nodes when there are degeneracy of eigenvalues (please refer to Berglund for details). At this juncture, it is appropriate to mention the related concepts of 
such that
for 
is negative showing 
, a neigbourhood of hyperbolic equilibrium point 
;
.
, 
can be combined as
will still belong to the set. Should be stated here that the fact that we have the linear addition of points necessitates the space it belongs to is a linear space (hence the use of 
). One could in fact remove the requirement of a special point (say, the origin) in vector spaces and consider the embedding space being simply an affine space with linear addition operation still in place.
and that the convex combination involves a real parameter 
. One question comes to mind is how this generalizes to complex linear spaces (as in quantum theory). Found a paper, “
and embed the unit 
to form 
in the convex combination. This will imply the loss of ordering. However they also mention the use of inner product of elements of the embedding space to form the necessary reals if required.![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=000000&s=0&c=20201002)
and it obeys additivity property
,
. One can extend this to continuous labelled events and the sum extends to an integral. Generalizing even further one can let the measure be defined for (Minkowski) sum of convex sets (say, 
) in such a way that they obey 
.
.
–convex measure. The special case of 
gives
.
(typically density matrices if the Hilbert space is finite-dimensional). Generally density operators 
;
;
.
(twice projection has the same effect as the first single one).
. Hence the relation of convex geometry to quantum theory. While mixed states are the more general form (which includes pure ones), it is normally the case that one needs to define what the pure states are first (through some procedure, say, quantization). The other interesting point is that the expression of the mixed state in terms of pure state is not unique; physically translating as two statistical mixtures with same statistical averages cannot be distinguished. Mielnik likened this to be similar to the use of local coordinates in Riemannian geometry and promoted convex geometry as the geometry for (even a generalised) quantum mechanics – see Mielnik, “
and 
. A subset 
,
lie in 
-face is a facet. Given the set of all feces of a convex body, one can form a partial ordering i.e. 
if face 
is contained in 
. This leads to another idea: partial ordering is the essential ingredient to form a logic, allowing what statements can be made. Thus convex geometry is related to the idea of quantum logic proposed much earlier and this is discussed in Mielnik, “
obeying Maxwell’s equations. It is convenient to introduce scalar and vector potentials 
to help solve such that
i.e.
are defined modulo phase factors (see B. Felsager, 
but yet retains the canonical commutation relations implying a form of gauge symmetry. See
where 
is the symplectic potential often given as 
(see P. Libermann & C.-M. Marle, 
.
. (see Ciaran Hughes, “A Brief Discussion on Gauge Theories” for a brief introduction, 
giving Klein-Gordon equation 
.
giving Dirac Equation 
.
giving Maxwell equation 
.
, one consider the transformations by a (global) phase factor:
.
group element. The free fermion Lagrangian above is invariant under this global phase transformation.
with the Lagrangian
.
transformations
.
sending 
to 
. Then the Lagrangian transforms as
such that the derivative transforms as needed:
.
,
is an element of 
is the (gauge) coupling constant. Note that the ambiguity of the gauge fields (potentials) is still there with the gauge field transforms as
.
(summed over generators of the Lie algebra 
. This is given via the field strength tensor![F_{\mu\nu}= F^a_{\mu\nu} T^a=[D_\mu,D_\nu]](https://s0.wp.com/latex.php?latex=F_%7B%5Cmu%5Cnu%7D%3D+F%5Ea_%7B%5Cmu%5Cnu%7D+T%5Ea%3D%5BD_%5Cmu%2CD_%5Cnu%5D&bg=ffffff&fg=000000&s=0&c=20201002)
.
and that
.
as such term will break the gauge invariance of the Lagrangian.Hence at this stage, gauge fields are considered massless.
gauge theory. The complication is that the gauge fields that mediate the weak interaction are massive (the 
particles) and hence must include a further ingredient which is the Higgs field and the idea of spontaneous symmetry breaking, which is the subject of the conference mentioned earlier. What one sees as weak interaction and electromagnetic interaction are really remnants of the 
. See J. Hucks, “
which is understood as the square root of the Laplacian 
i.e. 
. If it is one dimensional with 
, this would have been straightforward. For more than one dimension, this is nontrivial – what is the square root of say 
?
-deformed numbers
are integers. By making 
, one then has the alpha-deformed addition
. For more cases of the integer, it is
i.e. it works like the ordinary derivative with numbers replace by the deformed numbers. One could generalise this further to power series functions. For instance the 
and form the 
where 
is the current density of the loop. The current may be carried by a charged particle orbiting in a loop. From the formula given, the magnetic moment is then proportional to its angular momentum.
, then its magnetic moment is
is the Bohr magneton and 
is the gyromagnetic factor (dimensionless magnetic moment). Now according to Dirac’s theory, 
but this quantity receives corrections from QFT which can be written as
can be written as a (divergent) power series of the fine structure constant 
Ketchup Spills 