(no exposition on their connection here justa juxtaposition)
Note: in the flow of this discussion there is a little ambiguity which is rather clarified to the end. If you are uncomfortable with this drop a line.
More than a week ago my friend Synch Sample asked me a few things, I usually get back sooner if I am not completely taken over by something else. Here is my response following his queries. Interesting, read if you wish. (in my physics idiosyncrasies you are allowed to read that like: interesting read, if you wish)
SAMPLE: Hello Dash – I hope all is good and well with you. You’d put up a post on the full moon. My smart phone has an app running in the background that notifies me of the phases of the moon & indeed last night was a full moon. I am curious to know though does a full moon in California necessarily translate to a full moon 12 times zones away on the other side of the globe. As I’m aware you’re a jet setter, so I never know your location at any one time.
The question I had Dash is a trivial one, but one that I have never gotten a clear answer to. Years ago, around 1991 I was sitting in a dark alley across from a Winchell’s Donuts. I saw what looked like a manual lying on the ground by some dumpsters & curiosity got the best of me. I picked up this manual & started browsing through it. Low & behold it was a course manual for UCLA. Naturally I started flipping through it & went directly to the mathematics & physics sectiions. UCLA at the time was extremely strong in logic & they offered many courses in that area. Basically I’d say there where over hmm 100 courses in both pure & applied mathematics in that manual. Pure being group theory, Algebras, set theory, number theory, euclidean & algebraic geometry as well has discrete & concrete mathematics. In applied mathematics you had courses with headings such as fluid dynamics, mechanics, partial differential equations – courses 1,2,3,4 & 5, advanced partial diff equations., complex variables..etc. There where also other applied mathematics courses – these were under the heading gauge theory, perturbation theory, statistical mechanics, Riemann Geometry & Brownian Motion – other course offerings included Spinors, Tensors & Twistors. When I flipped to the physics section I saw most of the courses where related to solid state physics, the others with QM as well as field theory, gravitation, Relativity & optics – others where Thermodynamics & Heat transfer as well as EM. Outside of the Gravitation, field theory & relativity I saw no courses on theoretical physics.
So I am confused here – why was Gauge Theory, Riemann Geometry, Statistical Mechanics, Brownian Motion & Perturbation theory part of the mathematics course curriculum and taught in the mathematics department & not part of the physics course curriculum.
Last thing, in plain English can you explain what Gauge Theory is, is it’s function to describe the weak interactions in physics – when the term SU2 is used what the hell does this mean – last thing – what I was unaware of is that the e – charge is a composite of 2 types of charge. I was told A decomposition of a group under U(1)^k gives you all electric charges you need to know again u – is that Unitary Group I’m confused –
DSAH: will get back to you later
Today: I would say yes, the full moon would be also visible when your evening sets in, but the size of the moon may not be the same. The moon is fully visible when it comes between sun and earth so time zone would not affect the fullness, it just affects the size. (I haven’t reviewed anything but saying at the top of my mind)
To answer your 2nd question, the course curriculum is a matter of how the departments design them to be, if there is plenty of maths and the maths department has active physics research in theory of physics they may have such arrangements.
A gauge theory basically deals with unification of fundamental forces and symmetry using gauge functions, as far as I know. The symmetry is a transformation on the force equations eg the Maxwell equations. (also called wave equations) The electric and magnetic fields are described in terms of a scalar function and a vector function (a vector function or field is a 3 dimensional function, Fx, Fy and Fz) SO if you add some well chosen functions to these scalar and vector functions it will change the scalar and vector functions or fields but not change the resulting electric and magnetic field.
This is called a gauge symmetry and gauge transformation since the additional functions are called gauge functions. This helps tremenedously since this unifies electric and magnetic fields into just one force field called the electromagnetic field. This then helps in theory of Relativity and theory of relativity is a unification of electric and magnetic fields in the same way in relativity space and time are unificated. (or unified) Now these gauge functions which has such nice behavior are categorized into different groups. Groups are mathematical structures much like vectors and tensors. These are like matrix whose transformation properties define what is called unitarity and determinants. A SU2 is the acronym for special unitary group. If I remember, at the top of the mind: special stands for determinant 1, and unitary stands for: matrix*matrix_hermitian_conjugate = unit matrix. If the matrix is a order/dimension two its called group-2. Similarly a one-dim matrix or order 1 matrix is called U(1) group. Different kind of forces exihibit different group properties in their gauge functions.
Note: probabilities of different processes in our advance theories occur as elements of these Unitary matrix. They have to be unitary since they constitute what is called a basis states. A basis states is like what you have in sound-physics (acoustics) where each mode of vibration is a state representing different energy. In quantum mechanics the basis states are also called eigenstates or wavefunction states whose expectation values are the physical quantities: energy, momentum etc. SO in a scattering process let’s say neutrinos being produced from mesons and these mesons being prodduced from protons the quantum mechanical wavefunctions or simply called state-vectors are arranged in their column matrices and the operations of parity and other force field operations can be achieved through other matrices categorized as U1 or SU2.
When you solve eg Schrodinger wave equation which can be done in various formalisms the elements of these groups or matrices form different products each giving various probabilities of these processes, eg what will be the branching rates of the neutrinos, what is the total number of protons and how many mesons will be produced and so on. SO the forces or energy which are represented through Hamiltonians/Lagrangians act on these wavefunction states represented through these other matrices and we get results for energy and and other physical quantities …
Note: Hamiltonians are matrices themselves which represent what are called quantum mechanical operators such as “i*h-cross*d/dx” and “x-cap”. The wavefunction states or state vectors are row or column matrices having each eigen-state as an element. And Su2 or U1 are matrices that represent transformation operations, eg going from one type of force to another … A parity operation would be achieved through these gauge groups.