main article: inculcation

I make a very specific use of the words “physics” and “mechanics”, here. Of course I shall explain what I mean, but do note, it is a personal choice.

all of physics, all at once

Before doing anything, just watch this:

Mechanics using the Action principle

(ignore the Fourier expansion of electric field stuff, just the ideas behind GR, QFT matters!)

Continue with GR and action: General Relativity by Prof. Thanu Padmanabhan - YouTube

After that watch these lectures covering Newtonian, Lagrangian, Hamiltonian, Statistical mechanics, special relativity all at once!

These are nice as an intro “proper” physics, they will look fascinating, but my recommended levels of motivation and precision is absent. These are “Feynman lectures done right”. One may watch his non-linear dynamics and quantum mechanics lectures right after this.

But nothing is explained rigorously, although hinted at, lots of details are skipped and Balki name drops a LOT of stuff. You may choose to ignore them initially, because each term becomes a rabbit hole for math topics.

Balakrishnan's quantum physics

what is physics actually then

From these previous lectures, one must agree that doing physics is a three step process:

Step 1: Choose your spacetime, you have 3 major types of options:
- $c^{- 1} = 0$ Newtonian: the good ol’ “non-relativistic” spacetime
- $c^{- 1} = 1, G = 0$ Minkowski: the SR spacetime
- $c^{- 1} = 1, G = 1$ Lorentzian: a whole range of GR spacetimes
Step 2: Choose what the contents of your spacetime must be:
- points: point particles, bodies with finite number of degrees of freedom, rays
- fields: infinite degrees of freedom
- fluids: “spacetime itself flowing” (they are different from fields, yes)
- condensed matter
Step 3: Choose a description for the contents (although not all of the following is possible for all the contents in step 2)
- $ℏ = 0$ “classical mechanics/classical field theory” description - that is, writing equations of motion, $ma = F$ for particles in $c \to \infty$ or Maxwell’s equations for EM fields
- $ℏ = 0$ “classical statistics” description
- $ℏ = 1$ “quantum mechanics/QFT” description
- $ℏ = 1$ “quantum statistical” description

BUT! Does the description of matter really depend very much on the spacetime? Yes sure, the equation of motion will change drastically, but the methods in ODEs do not change at all! We can study about spherical harmonics in waves, electrodynamics and in QM class - their physical interpretation is different but their math interpretation remains the same!

Hence, there is a component to this in the second step of the 3 step process: which is “abstract”. How much of the things we study can we abstract out? Can it be really useful in doing physics?

This makes sense to me because generally quantum mechanics is taught before fluid mechanics, which is true because the former is a linear PDE - which makes it much easier - just introduce eigenvectors and eigenvalues and the method can be explained!

Now for an example:

Around 1850 Maxwell realized that the field strength of the electromagnetic field is modeled by what today we call a closed differential 2-form on spacetime. In the 1930s Dirac observed that more precisely this 2-form is the curvature 2-form of a U(1)-principal bundle with connection, hence that the electromagnetic field is modeled by what today is called a degree 2-cocycle in ordinary differential cohomology . ^[https://ncatlab.org/nlab/show/higher+category+theory+and+physics#GaugeTheory]

This is an example of the table for electromagnetic field (for $ℏ = 0$ ). Gauge fields will be discussed in Doing the proper theory of gauge fields.

Although we must do physics in the more specific sense (in contrast to abstract sense) as well, so we make this little table for reference with common terminology:

contents	description	$c^{- 1} = 0$ Newtonian spacetime	$c^{- 1} = 1, G = 0$ Minkowski spacetime (SR)	$c^{- 1} = 1, G = 1$ Lorentzian manifolds (GR)
	$ℏ = 0$
points	“actual”	“classical mechanics” Kleppner, Goldstein, David Morin, LandauLifshitz vol 1, Balki’s lectures	“relativistic mechanics” LandauLifshitz vol 2	“general relativity” LandauLifshitz vol 2
	“stat”	“statistical mechanics” Kardar	“relativistic statistical mechanics” Palash Pal
fields	“actual”	“non-relativistic classical field theory”	“classical field theory”	“classical field theory in curved spacetime”
	“stat”		“statistical field theory”
fluids		“fluid mechanics”	“relativistic fluid mechanics”	“fluid mechanics in curved spacetime”

	$ℏ = 1$
points	“actual”	“quantum mechanics” Griffiths, Balki’s lectures, Shankar	-	-
	“stat”	“quantum statistical mechanics” QM textbooks ^ should cover this	-	-
fields	“actual”		“QFT”	“QFT in curved spacetime” or “global QFT”
	“stat”
fluids	quantum fluids?	?	?	?
atoms?
condensed matter?
				QG?

In general, there are

Landau Lifshitz volumes 1-10
David Tong’s notes for many of the elements in the table
look below for more!

So, essentially we did all of physics together. But did we do it properly? Did we went onto understanding the details?

How to do mechanics properly

understanding the philosophy

How one should do is by constructing everything linearly.

Schuller’s lectures are a good place to start:

Lectures on Geometrical Anatomy of Theoretical Physics by Frederic Schuller

These lectures start from logic!

But for more, after doing inculcation-analysis-finite-vector-spaces and inculcation-smooth-manifolds we can do:

inculcation-odes $\to$ Mechanics of points done properly
functional analysis, PDEs $\to$ QM, fluids, etc

Doing ODEs with proper rigor helps eliminate errors from intuition, cyclic reasoning etc. this is the whole reason why inculcation-linear-constructions exists!

But there are a whole lot of benefits of abstract thinking and relying on general framework as well.

Philosphy: ODEs do not care about spacetime or coordinates or other structures

For instance, what one must understand is (among many other things), in the side of ODEs: Lagrangian, Hamiltonian etc really do not care about the spacetime, or what you are trying to describe even: give it a ray of light in Newtonian spcaetime, or a point particle moving around a black hole (Swartzchild spacetime): the description remains the same.

Configuration space $\leftrightarrow$ Lagrangian dynamics, any other ODE

Phase space $\leftrightarrow$ Hamiltonian dynamics

I do not mean the equations remains the same btw! I just mean use can use the general prescription (math!) like Lagrangians and Hamiltonians for any spacetime!

This would therefore mean, the “statistical mechanics” prescription would also be, in this way - because statistical mechanics just starts from the phase space - the methods independent of “spacetimes” per say as phase space only depend on the configuration spaces.

doing the spacetimes bit

This doesn’t need to be done first, or before Mechanics of points done properly, but these lectures are amazing:

The WE-Heraeus International Winter School on Gravity and Light

THE WE-HERAEUS INTERNATIONAL WINTER SCHOOL ON GRAVITY AND LIGHT (Spanish Translation Available!) (richie291.wixsite.com)

Maths with Physics: The WE-Heraeus International Winter School on Gravity and Light, Lectures, Tutorials and Solutions

Mechanics of points done properly

That is: as we see in “classical mechanics”, but the idea of a configuration space captures rays (as in ray optics) and rigid bodies along with finite number of point particles.

writing the equations

	Configuration space	$V = 0$ solutions
1 point	$R^{3}$ (space)	geodesics in $R^{3}$ with usual metric, that is, straight lines
$n$ points	$(R^{3})^{n}$
1 rigid body	$SO (3) ⋉ R^{3}$ (a Lie group!)

The famous textbooks are

V. I. Arnold - Mathematical Methods of Classical Mechanics-
Abraham R., Marsden J.E. - Foundations of Mechanics (1987)
Jerrold E. Marsden, Tudor S. Ratiu - Introduction To Mechanics And Symmetry A Basic Exposition of Classical Mechanical Systems-Springer (2010)

Similar tables can be found here:

interpreting a lot of systems as geodesic flows and vice versa

manifold metric system corresponding to geodesics
configuration space $Q$ be any manifold $g_{ij}$ coming from a (non deg) quadratic Lagrangian $g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j}$ (with no potential term) Lagrangian dynamics for $L = g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j}$
a surface in $R^{3}$ induced by the metric in $R^{3}$ force free (Newtonian) motion of the particle constrained on the surface, geodesic equation is literally ” $0 \frac{F}{m} = a$ ” here
$SO (3)$ (configuration space of rigid body with one fixed point, say center of mass) given by the torque-free Lagrangian of a rigid body torque-free motion of the rigid body AKA Euler top
space $Q$ which can be $R^{n}$ or any manifold $n (q)^{2} δ_{ij}$ where $n : Q \to (0, \infty)$ is refractive index light rays in optics
configuration space $Q$ with metric $g$ $h_{ij} = \frac{2}{m} (E - V (q)) g_{ij}$ particle in potential $V$ in space $(Q, g)$ with energy $E$
GR spacetime $M$ which can be any manifold a metric $g$ with Lorenzian signature (relativistic) free particle in the spacetime or light trajectories if it’s a null geodesic
spacetime $(M, g)$ cross $U (1)$ that is $M \times U (1)$ $h = [h_{ij} h_{i θ} h_{θ i} h_{θθ}] = [g_{ij} + A_{i} A_{j} - A_{i} - A_{i} 1]$ Kaluz-Klein theory of particle of charge $q$ on a GR spacetime with electromagnetic field

^[Classical Mechanics]

More examples where the configuration space is specifically a group can be found in

Table of configuration spaces that are groups a metric and geodesic flows on them

Lecture1 slides of Geometric Fluid Dynamics, Fall 2021

Link to original

manifold	metric	system corresponding to geodesics
configuration space $Q$ be any manifold	$g_{ij}$ coming from a (non deg) quadratic Lagrangian $g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j}$ (with no potential term)	Lagrangian dynamics for $L = g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j}$
a surface in $R^{3}$	induced by the metric in $R^{3}$	force free (Newtonian) motion of the particle constrained on the surface, geodesic equation is literally ” $0 \frac{F}{m} = a$ ” here
$SO (3)$ (configuration space of rigid body with one fixed point, say center of mass)	given by the torque-free Lagrangian of a rigid body	torque-free motion of the rigid body AKA Euler top
space $Q$ which can be $R^{n}$ or any manifold	$n (q)^{2} δ_{ij}$ where $n : Q \to (0, \infty)$ is refractive index	light rays in optics
configuration space $Q$ with metric $g$	$h_{ij} = \frac{2}{m} (E - V (q)) g_{ij}$	particle in potential $V$ in space $(Q, g)$ with energy $E$
GR spacetime $M$ which can be any manifold	a metric $g$ with Lorenzian signature	(relativistic) free particle in the spacetime or light trajectories if it’s a null geodesic
spacetime $(M, g)$ cross $U (1)$ that is $M \times U (1)$	$h = [h_{ij} h_{i θ} h_{θ i} h_{θθ}] = [g_{ij} + A_{i} A_{j} - A_{i} - A_{i} 1]$	Kaluz-Klein theory of particle of charge $q$ on a GR spacetime with electromagnetic field

Here there are PDE dynamics on a infinite dynamical space thought of as “mechanics of a point”! Yes, we can think of fluid dynamics as classical dynamics on an infinite dimensional phase space! This phase space is not a vector space as the equation is non-linear.

solving the equations

Physics textbooks on “classical mechanics” aren’t rigorous, they don’t worry about a lot of things. What exactly are we missing then?

only worry about local properties of the configuration spaces, local solutions of the equations: so for example it cannot differentiate between a cylinder or a sphere as configuration spaces, because locally they are “same” given how we are describing them (smooth manifolds).
both local and global properties of ODEs are studied in inculcation-odes, in for example Perko’s book.

geometry behind the dynamics

The geometry behind dynamics of a ODE is that of a vector field on a manifold.

vector fields and ODE dictionary

What we do is, write a differential equation like
$\overset{x}{˙}_{i} = f_{i} (x_{1}, x_{2}, \dots, x_{n})$
for all $1 \leq i \leq n$ so we have $n$ equations for $n$ variables $x_{i}$ and make it even more compact by
$\dot{x} = f (x)$
where $f : U \subseteq R^{n} \to R^{n}$ is a vector field on the open domain $U$ .

This gives us a geometric pov on ODEs in $R^{n}$ , and we have a

solving differential equations analysis and geometry of vector fields
an equation $\overset{x}{˙} = f (x)$ a vector field $f : U \subseteq R^{n} \to R^{n}$
solutions of the equation $\overset{x}{˙} (t) x (0) = f (x (t)) = x_{0}$ integral curves of the vector field $t \mapsto Φ_{t}^{f} (x_{0})$
how solutions depend on initial conditions $x_{0} \mapsto x (t)$ flows of the vector field $x_{0} \mapsto Φ_{t}^{f} (x_{0})$
conserved quantities integrals of the vector field
(linearly) decoupling the differential equation $\overset{z_{i}}{˙} = f_{i} (z_{i})$ (linear) coordinate transformation such that $f = \sum_{i} f_{i} (z_{i}) \overset{z}{^}_{i}$

This is a standard geometric interpretation.

We may convert ordinary differential equation of any order to first order by taking enough independent variables and defining them to be higher derivatives.
Link to original

solving differential equations	analysis and geometry of vector fields
an equation $\overset{x}{˙} = f (x)$	a vector field $f : U \subseteq R^{n} \to R^{n}$
solutions of the equation $\overset{x}{˙} (t) x (0) = f (x (t)) = x_{0}$	integral curves of the vector field $t \mapsto Φ_{t}^{f} (x_{0})$
how solutions depend on initial conditions $x_{0} \mapsto x (t)$	flows of the vector field $x_{0} \mapsto Φ_{t}^{f} (x_{0})$
conserved quantities	integrals of the vector field
(linearly) decoupling the differential equation $\overset{z_{i}}{˙} = f_{i} (z_{i})$	(linear) coordinate transformation such that $f = \sum_{i} f_{i} (z_{i}) \overset{z}{^}_{i}$

The geometry of a Lagrangian is behind the algebra of a chain complex called “variational bicomplex” that it creates, (apart from the optimization problem). I’ve not read much about it.
We can even convert some Lagrangian problems to geodesics on some spaces which becomes Riemannian geometry. Exploring more on this here: inculcation-geodesic-flows
The geometry behind Hamiltonian dynamics is studied under the name of symplectic geometry. A good intro reference with regards to mechanics is: https://people.math.harvard.edu/~jeffs/SymplecticNotes.pdf

probabilistic description of points

As done in statistical mechanics.

Main goal of statistical mechanics

derive the empirical laws of thermodynamics from the classical mechanics description (so just ” $F = ma$ “)

Boltzmann, Gibbs invented statistical mechanics to give more meaning to the thermodynamic quantities
Ergodic theory was invented to ask when does the assumptions made by Boltzmann and others hold.
My first reference was Balki’s lectures on classical physics, other physics references include MIT 8.333 Statistical Mechanics I: Statistical Mechanics of Particles, Fall 2013 - YouTube
Statistical Mechanics From Thermodynamics to the Renormalization Group
Roderich Tumulka’s notes is best reference I’ve seen, actually talks about the main goal
Information Geometry (ucr.edu)
- Part 17-21 is where thermodynamics is discussed
Classical Mechanics versus Thermodynamics (ucr.edu)

From John Baez’s ideas we what we get is

Probabilistic : CM (X, μ) \to StatMech (Q, π) \to Thermo (Q, S)

So did we succeed? Did we (does the above references) “prove” thermodynamics from just ” $F = ma$ “?

No. Not equilibrium statistical mechanics anyways. The following are some pitfalls:

Bug

People claim “Luoville’s theorem of Hamiltonian dynamics explains equilibrium statistical mechanics” $\leftarrow$ this makes no sense! Luoville’s theorem just says the vector field corresponding to the ODE has divergence 0 which is $⟺$ its flow is volume preserving. Bazzilion many ODEs (even physical ones!) preserve volume but they don’t even come close to having properties like thermalization.

Dynamics of $n$ harmonic oscillators is such an example which is even PERIODIC in time (for specific parameters)! hah! People even compute $Z (β) = \int_{R^{2 n}} exp (- β H)$ for this, please give me ANY physical interpretation of this computation!?!

If you think “ergodicity” of the flow is enough assumption to reproduce equilibrium stat mech look at ‘ergodicity, as usually stated, is neither sufficient not necessary for thermalization’

When Poincaré recurrence theorem is used to contradict existence of a thermal equilibrium, people are quick to say “the recurrence times are LARGER compared to thermalization times” $\leftarrow$ which is true until it isn’t!

Fermi and company found this curious case experimentally: Fermi–Pasta–Ulam–Tsingou problem - Wikipedia

You can’t change information of $1 0^{23}$ dimensions into $4$ variables, sadly, nope. How is thermodynamics true then? In what sense I mean? Well…Roderich Tumulka’s notes above tries to explain a lot. I oscillate between being convinced and not.

However, the following book has a “proof” of Boltzmann’s (non-equilibrium stat mech) equation for dilute gases in some very precise sense.

Carlo Cercignani, Reinhard Illner, Mario Pulvirenti - The Mathematical Theory of Dilute Gases-Springer-Verlag New York (1994)

quantization of the mechanics of points

As done in quantum mechanics. Essentially, functional analysis on (rigged) Hilbert spaces (brings in representation theory)

Woit’s Quantum Theory, Groups and Representations
Brian C. Hall - Quantum Theory for Mathematicians-Springer-Verlag New York (2013)
Leon A. Takhtajan - Quantum Mechanics for Mathematicians-American Mathematical Society (2008)

Frederic Schuller's lectures on quantum mechanics with lecture notes

Maths with Physics: Frederic Schuller’s Lectures on Quantum Theory with Lecture Notes

DR. FREDERIC SCHULLER’S COURSE OF QUANTUM THEORY (richie291.wixsite.com) e So let’s ponder on what quantum mechanics does really. It gives us a linear dynamics from a ODE:

Quantizaton : CM (X, H) \to QM (L^{2} (X), \hat{H})

but now we observe the correspondence between them, even though they are very different structure wise:

though the one point problem

We must notice the analogy

the “classical” problem in Hamiltonian dynamics the quantum problem in Schrodinger/H picture
(physical) space is $R^{3}$ and time is $R$ (physical) space is $R^{3}$ and time is $R$
configucation space is set of all positions possible $R^{3}$ the domain of the wave function is $R^{3}$ (= configuration space)
the phase space is set of all positions, momentum pairs possible $R^{3} \times R^{3}$ whose “dimension” is $6$ the ”quantum phase space” is the set of all wave functions, set of all square integrable functions $R^{3} \to \C$ $L^{2} (R^{3}, \C)$ whose “dimension” is not finite!
Hamiltonian is a smooth function $R^{3} \times R^{3} \to R$ $H (q, p) = \frac{p ^{2}}{2 m} + V (q)$ the “Hamiltonian” is a Hermitian linear functions $L^{2} (R^{3}) \to L^{2} (R^{3})$ $\hat{H} = \frac{p ^ ^{2}}{2 m} + \hat{V}$
Hamilton’s equations Schrodinger equation
Observables are smooth functions $R^{3} \to R$ (just like Hamiltonian) Observables are Hermitian linear functions $L^{2} (R^{3}) \to L^{2} (R^{3})$ (just like quantum “Hamiltonian”)
The set of all observables is the set of all smooth functions, so $C^{\infty} (R^{6})$ where we can define the Poisson brackets ${,}$ The set of all observables is the set of all Hermitian linear functions $L^{2} (R^{3}) \to L^{2} (R^{3})$ , this is a subspace of all linear functions and is closed in the commutator brackets $[,]$
position $q_{i}$ $\overset{q}{^}_{i}$
momentum $p_{i}$ $\overset{p}{^}_{i} = \pdvf q_{i}$
momentum “generates” translations - $exp (α X_{p_{i}})$ ( $X_{p_{i}}$ is the Hamiltonian vector field of $p_{i}$ , exponential of vector fields) - translation is a group action by $(R^{3}, +)$ onto the phase space $R^{6}$ - which is not a symmetry here! because ${H, p_{i}} \neq = 0$ if we have a non-constant potential momentum “generates” translations - $exp (α \overset{p}{^}_{i})$ (exponential of linear maps) - translation is a group action by $(R^{3}, +)$ onto the phase space $L^{2} (R^{3})$ - which is not a symmetry here! because $[H, p_{i}] \neq = 0$ if we have a non-constant potential
$L_{i}$ $\hat{L}_{i}$
angular momentum “generates” rotations - $exp (θ X_{L_{i}})$ ( $X_{L_{i}}$ is the Hamiltonian vector field of $L_{i}$ , exponential of vector fields) - rotation is a group action by $SO (3)$ onto the phase space $R^{6}$ - which is a symmetry here! because ${H, L_{i}} = 0$ if we have a non-constant potential angular momentum “generates” rotations - $exp (θ \hat{L}_{i})$ (exponential of linear maps) - rotation is a group action by $SO (3)$ onto the quantum phase space $L^{2} (R^{3})$ - which is a symmetry here! because $[H, L_{i}] = 0$ if we have a non-constant potential
the group action is symplectic, and more specifically Hamiltonian! the above group actions act by unitary transformations because exponential of Hermitian maps are unitary
${f_{1}, f_{2}} = 0$ means we are one step closer to Liouville–Arnold integrability $[\hat{f}_{1}, \hat{f}_{2}] = 0$ means we can simultaneously diagonalize them $\to$ one step closer to having an eigenbasis (is this actually correct?)

This can be said in brief by saying its a Lie algebra homomorphism from a Lie subalgebra $A$ of observables on (classical) phase space $P$
$A \subseteq (C^{\infty} (P), {}) \to (End (H), [,])$
This is precisely what quantization is!

This however, might not be unique, or even exist for any set $A$ .

This analogy can be generalised many folds (geometric quantization of a Hamiltonian group action).
Link to original

the “classical” problem in Hamiltonian dynamics	the quantum problem in Schrodinger/H picture
(physical) space is $R^{3}$ and time is $R$	(physical) space is $R^{3}$ and time is $R$
configucation space is set of all positions possible $R^{3}$	the domain of the wave function is $R^{3}$ (= configuration space)
the phase space is set of all positions, momentum pairs possible $R^{3} \times R^{3}$ whose “dimension” is $6$	the ”quantum phase space” is the set of all wave functions, set of all square integrable functions $R^{3} \to \C$ $L^{2} (R^{3}, \C)$ whose “dimension” is not finite!
Hamiltonian is a smooth function $R^{3} \times R^{3} \to R$ $H (q, p) = \frac{p ^{2}}{2 m} + V (q)$	the “Hamiltonian” is a Hermitian linear functions $L^{2} (R^{3}) \to L^{2} (R^{3})$ $\hat{H} = \frac{p ^ ^{2}}{2 m} + \hat{V}$
Hamilton’s equations	Schrodinger equation

Observables are smooth functions $R^{3} \to R$ (just like Hamiltonian)	Observables are Hermitian linear functions $L^{2} (R^{3}) \to L^{2} (R^{3})$ (just like quantum “Hamiltonian”)
The set of all observables is the set of all smooth functions, so $C^{\infty} (R^{6})$ where we can define the Poisson brackets ${,}$	The set of all observables is the set of all Hermitian linear functions $L^{2} (R^{3}) \to L^{2} (R^{3})$ , this is a subspace of all linear functions and is closed in the commutator brackets $[,]$
position $q_{i}$	$\overset{q}{^}_{i}$
momentum $p_{i}$	$\overset{p}{^}_{i} = \pdvf q_{i}$
momentum “generates” translations - $exp (α X_{p_{i}})$ ( $X_{p_{i}}$ is the Hamiltonian vector field of $p_{i}$ , exponential of vector fields) - translation is a group action by $(R^{3}, +)$ onto the phase space $R^{6}$ - which is not a symmetry here! because ${H, p_{i}} \neq = 0$ if we have a non-constant potential	momentum “generates” translations - $exp (α \overset{p}{^}_{i})$ (exponential of linear maps) - translation is a group action by $(R^{3}, +)$ onto the phase space $L^{2} (R^{3})$ - which is not a symmetry here! because $[H, p_{i}] \neq = 0$ if we have a non-constant potential
$L_{i}$	$\hat{L}_{i}$
angular momentum “generates” rotations - $exp (θ X_{L_{i}})$ ( $X_{L_{i}}$ is the Hamiltonian vector field of $L_{i}$ , exponential of vector fields) - rotation is a group action by $SO (3)$ onto the phase space $R^{6}$ - which is a symmetry here! because ${H, L_{i}} = 0$ if we have a non-constant potential	angular momentum “generates” rotations - $exp (θ \hat{L}_{i})$ (exponential of linear maps) - rotation is a group action by $SO (3)$ onto the quantum phase space $L^{2} (R^{3})$ - which is a symmetry here! because $[H, L_{i}] = 0$ if we have a non-constant potential
the group action is symplectic, and more specifically Hamiltonian!	the above group actions act by unitary transformations because exponential of Hermitian maps are unitary
${f_{1}, f_{2}} = 0$ means we are one step closer to Liouville–Arnold integrability	$[\hat{f}_{1}, \hat{f}_{2}] = 0$ means we can simultaneously diagonalize them $\to$ one step closer to having an eigenbasis (is this actually correct?)

Mechanics of fields

🚧 This section is still under construction.

Why upgrade to Lagrangian fields?

“Classical field theory” as a course was COOKED up to teach QFT because any quantum theory is defined by a classical theory and then we quantize it

quantization: ClassicalFT → QFT

Doing the proper theory of Lagrangian fields

so the actual content must be this

given you know vector bundles on smooth manifolds, we say field configuration bundles $=$ some vector bundle on spacetime
this is what i call spacetime fields , now write a Lagrangian and study it, the spacetime be a semi-Riemannian manifold (not flat), then div of stress energy tensor= 0 doesn’t imply global conservation laws, we must have Killing vector fields
you might have use the representation $Sp in (n) \to SO (n)$ , which allows you to define spin bundle where $Sp in (n)$ bundles

spin geometry $:=$ study of dirac operators on such bundles

Doing the proper theory of gauge fields

gauge fields DO NOT happen on spacetime! (that is, the field’s domain isn’t the spacetime manifold)

http://nicf.net/articles/classical-em/

The following books are okay:

Gregory L. Naber - Topology, Geometry and Gauge fields - two volumes
Mikio Nakahara - Geometry, topology, and physics

quantization of field theories

Mechanics of fluids

The physics of fluid mechanics: Averaging a differential equation system with high number of dynamical variables

Hamiltonian system of a system of particles
$↓$ Statistical mechanics of the system of particles: Boltzmann equation with the probability distribution $f (x, t)$ $↓$ Take the average of quantities at the point $(x, t)$ and get equations for velocity and pressure $v (x, t), p (x, t)$ :

Continuity of mass

Continuity of momentum

Continuity of energy

Entropy inequality

…is that it?

My question is can we do this for any Hamiltonian system? Or even any ODE with high enough ( $\sim 1 0^{23}$ ) dimension? Probably not. But let’s not worry about that!

There are various physical and computational aspects to it

Thus there is an analysis side to it.

But there is a geometric, and topological side to it too,

Turbulence is still an unsolved problem.

Mechanics with a lattice somewhere

Where does the Ising model sit in all these? Does it even need a “spacetime”?

R²

Explorer

Inculcation: all of mechanics

all of physics, all at once

what is physics actually then

How to do mechanics properly

understanding the philosophy

doing the spacetimes bit

Mechanics of points done properly

writing the equations

interpreting a lot of systems as geodesic flows and vice versa

solving the equations

geometry behind the dynamics

vector fields and ODE dictionary

probabilistic description of points

quantization of the mechanics of points

though the one point problem

Mechanics of fields

Why upgrade to Lagrangian fields?

Doing the proper theory of Lagrangian fields

Doing the proper theory of gauge fields

quantization of field theories

Mechanics of fluids

Mechanics with a lattice somewhere

Graph View

Table of Contents

Backlinks

R²

Explorer

Inculcation: all of mechanics

all of physics, all at once §

what is physics actually then §

How to do mechanics properly §

understanding the philosophy §

doing the spacetimes bit §

Mechanics of points done properly §

writing the equations §

interpreting a lot of systems as geodesic flows and vice versa §

solving the equations §

geometry behind the dynamics §

vector fields and ODE dictionary §

probabilistic description of points §

quantization of the mechanics of points §

though the one point problem §

Mechanics of fields §

Why upgrade to Lagrangian fields? §

Doing the proper theory of Lagrangian fields §

Doing the proper theory of gauge fields §

quantization of field theories §

Mechanics of fluids §

Mechanics with a lattice somewhere §

Graph View

Table of Contents

Backlinks

all of physics, all at once

what is physics actually then

How to do mechanics properly

understanding the philosophy

doing the spacetimes bit

Mechanics of points done properly

writing the equations

interpreting a lot of systems as geodesic flows and vice versa

solving the equations

geometry behind the dynamics

vector fields and ODE dictionary

probabilistic description of points

quantization of the mechanics of points

though the one point problem

Mechanics of fields

Why upgrade to Lagrangian fields?

Doing the proper theory of Lagrangian fields

Doing the proper theory of gauge fields

quantization of field theories

Mechanics of fluids

Mechanics with a lattice somewhere