## Solving Hilbert’s sixth problem (part three of many)

### A tale of two products

Thinking how to proceed on deriving quantum mechanics I came
to the conclusion to reverse a bit the order of presentation and review first
the goal of the derivation:

__a symmetric and a skew-symmetric product__.
Let us start with classical mechanics and good old fashion Newton ’s
second law: F = ma. Let’s consider the simplest case of a one dimensional
motion of a point particle in a potential V(x):

F = -dV/dx

a = dv/dt with “v” the velocity.

Introducing the Hamiltonian as the sum of kinetic and
potential energy: H(p,x)= p

^{2}/2m + V(x) we have:
dp/dt = - ∂V/∂x = -∂/∂x (p

^{2}/2m + V) = -∂H/∂x
and

dx/dt=v=∂/∂p(p

^{2}/2m + V) = ∂H/∂p
In general, one talks not of point particles, but it is
customary to introduce generalized coordinates

**q**and generalized momenta**p**to take advantage of various symmetries of the problem (**q**= q_{1}, q_{2},…,q_{n}**p**= p_{1}, p_{2},…,p_{n}).
The Hamilton’s equations of motion are:

d

**p**/dt = -∂H/∂**q**
d

**q**/dt = +∂H/∂**p**
with H = H(

**q**,**p**,t)
We observe two very important things right away:

__the equations are linear__and__q’s____and__**p’s**are in one-to-one correspondence**.**
{f,g} = ∂f/∂

**q**∂g/∂**p**- ∂f/∂**p**∂g/∂**q**
as a convenience to express the equations of motion like
this:

d

**p**/dt = {**p**, H}
d

**q**/dt = {**q**, H}
The Poisson bracket defines a skew-symmetric product
between any two functions f,g:

{f,g} = f o g = -g o f = -{g, f} and more importantly this
product obeys the so-called Jacoby identity:

{f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0

This identity follows identically from the definition of the
Poisson bracket and expanding and canceling the partial derivatives.

The Jacoby identity and the skew-symmetry property define a

__Lie algebra____.__
So in classical mechanics in phase space one defines two
products: the regular function multiplication: f(

**q**,**p**).g(**q**,**p**) which is**, and the Poisson bracket {f(**__a symmetric product__**q**,**p**),g(**q**,**p**)} which is__a skew-symmetric product.__**Now onto quantum mechanics**. In quantum mechanics one replaces the Poisson bracket with the commutator [A, B] = AB-BA which can be understood as

__a skew-symmetric product__between operators on a Hilbert space. There is also

__a symmetric product:__the

The commutator also obeys the Jacoby identity:

[A,[B,C]] + [C,[A,B]] + [B,[C,A]] =

[A, BC-CB] + [C, AB-BA] + [B, CA-AC]=

ABC-ACB –BCA+CBA + CAB-CBA -ABC+BAC
+BCA-BAC -CAB+ACB =0

and the commutator also defines

**, just like in classical mechanics.**__a Lie algebra__
How can we understand the Jordan
product? In quantum mechanics

__operators do not commute and we cannot simply take the function multiplication.__
To generate real spectra and positive probability predictions,
observable operators must be self-adjoint: O=O

^{†}meaning that in matrix form they are the same as the transposed and complex conjugate. However, because of transposition,__the product of two self-adjoint operators is not self-adjoint:__
(AB)

^{†}= B^{†}A^{†}=BA != AB
However, the Jordan
product preserves self-adjoint-ness:

{A,B}

^{†}= ½ ( (AB)^{ †}+ (BA)^{ †}) = ½ (BA + AB) = {A,B}
if A

^{† }=A and B^{† }=B
In quantum mechanics

**the****Jordan**

**product is a symmetric product.**

__Both classical and quantum mechanics have a symmetric and a skew-symmetric product:__
CM QM

Symm f.g Jordan
product

Skew-Symm Poisson bracket Commutator

__Both classical and quantum mechanics have dualities:__
CM: duality between

**q**s and**p**s:**q**<--->**p**
QM: duality between observables and generators:

**q**<---> -i ħ ∂/∂**q**=**p****So in this post we solved the simple direct problem: extract a symmetric and a skew symmetric product.**

In subsequent posts we will show two important things:

1) we
will derive the symmetric and skew-symmetric products of classical and quantum
mechanics from composability

2) we
will solve the inverse problem: derive classical and quantum mechanics from the
two products.

In the meantime: HAPPY NEW YEAR!

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In the meantime: HAPPY NEW YEAR!

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