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Strange "spin" is NOT a real thing

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- Derivation of Fermi-Dirac statistics.
- Perturbation theory and effective Hamiltonian.
- Derivation of Fermi golden rule.
- Green ( correlation ) function.

*(Eq.1)*

Ther **average** energy in temperature T can be gotten by multiplying them by Boltzmann factor ( β = 1/kT ).

In Eq.1, the energy (= ε ) is supposed to be a continuous value.

In actual phenomena, the energies are **quantized** by an integer times **de Broglie** waves.

Making energy ε **discrete**,

*(Eq.2)*

Substituting Eq.2 into Eq.1, we obtain

*(Eq.3)*

This is called "**Bose-Einstein** distribution".

*(Fig.1) Pauli exclusion principle is a force ? → "Shut up and calculate !"*

As you see in textbooks of the present condensed matter physics and spintronics, you will find these books are filled with only ( **unrealistic** ) mathematics, and they **NEVER** try to ask what spin (= Pauli exclusion principle ) really is.

These "Shut up and calculate !" attitudes of spintronics **obstruct** the development of science.

In "Pauli **exclusion** principle", two identical fermions **cannot** enter in the **same** place (= state ), even in very strong quarks.

This definition of the "same" place is very **vague**. How can we define these **boundary** lines ??

*(Eq.4)*

Fermi-Dirac statistics describes distribution of particles in certain systems comprising many identical particles that obey Pauli exclusion principle.

So the particle's number n is "**0**" or "**1**".

Like in Eq.1- Eq.3, we get **Fermi-Dirac** distribution in Eq.4.

Though this Pauli exclusion principle has the **greatest** power in all fundamental forces, these **boundary** lines inside which this principle is effective, are very **vague**. (= These are Only "**math**" and "symbol" worlds. )

*(Eq.5)*

In Eq.5, H_{0} and ε is the Hamiltonian and its energy in **free** particles.

"**V**" is some **interactive** Hamiltonian.

This V is usually complicated, so they use **approximate** Hamiltonian **H** like Eq.5.

*(Eq.6)*

The eigenvalues of this H (= **overall** Hamiltonian ) is "E", and eigenfunction is "Ψ", like Eq.6.

*(Eq.7)*

We expand Ψ and E in power of λ, as shown in Eq.7.

Ψ^{(0)} and E^{(0)} are equal to φ and ε of free particles.

Substituting Eq.5 and Eq.7 into Eq.6, we get

*(Eq.8)*

The left of Eq.8 means terms without λ (= zero order ).

We pick up terms which include λ to the power of **one** in Eq.8 right.

*(Eq.9)*

We define each Ψ is composed of (free) eigenfunctions φ of Eq.5, as shown in Eq.9.

And these φ are supposed to be **orthonormal**.

*(Eq.10)*

Using Eq.9 in Eq.8 right equation, and multiplying them by φ_{m} from the left side, we get Eq.10.

( Eq.5 is used, too. )

When "m" is not equal to "n", the last term of Eq.10 becomes zero.

*(Eq.11)*

Substituting Eq.11 into Eq.9, we get

*(Eq.12)*

Here "c_{n}" in Ψ^{(1)} is supposed to be zero.

Eq.12 is first-order perturbation.

*(Eq.13)*

In the same way, we pick up terms which include λ to the power of "2" in Eq.6 and Eq.7.

*(Eq.14)*

The first terms in both sides are the same, so they can be erased.

Substituting Eq.12 into Eq.14, we get.

*(Eq.15)*

As a result, we get

*(Eq.16)*

The first order perturbation of E (= second term of right side in the upper equation ) is gotten by multiplying Eq.10 by φ_{n} instead of φ_{m}.

We often use approximate energy in many-body system in condended matter physics.

So Eq.16 is important.

( This formula is only mathematical thing, which causes **divergence**. )

Fermi's golden rule means transition rate ( probability ) based on time-dependent Hamiltonian and perturbation.

*(Eq.17)*

Eq.17 means time independent Hamiltonian and energy of free particle system.

When Hamiltonian includes time dependent interactive part (= H'(t) ), Schrodinger equation becomes

*(Eq.18)*

Eq.18 is equal to

*(Eq.19)*

Multiplying Eq.19 by eigenstate "m" from left side,

*(Eq.20)*

Here we define

*(Eq.21)*

Substituting Eq.21 into Eq.20, and multiplying them by e^{iEt/h} from left side,

*(Eq.22)*

Integrating Eq.22 by t', multiplying them e^{-iEt/h}, and using the relation of Eq.21,

*(Eq.23)*

In Eq.23, we approximately change a_{n}(t') into A_{n}(0) in the right side.

And the following relations are used

*(Eq.24)*

Defining the initial state ( t= 0 ) as a simple form (= δ_{mn} ), and picking up only the second term (= interaction ) in the right side of Eq.23,

*(Eq.25)*

Multiplying Eq.25 by its complex conjugate, we get

*(Eq.26)*

Here we use the following relations,

*(Eq.27)*

Eq.26 becomes

*(Eq.28)*

Delta function and its formula are

*(Eq.29)*

And we define E_{mn}, as follows,

*(Eq.30)*

We use the following mathematical trick,

*(Eq.31)*

From Eq.28 to Eq.31, we get

*(Eq.32)*

In the first line of Eq.32, using the formula of delta function, "ħ" of E_{mn} inside delta function is erased.

Dividing Eq.32 by time "t", we can get transition rate (= Fermi golden rule ).

The problem is these kinds of transition probabilities are often **divergent** to infinity due to **delta** function.

So we should get away from "mathematical" world, and **redefine** various physical phenomena by "**real**" objects.

( The same thing can be said about **unrealistic** "singularity" in the mathematical black hole. )

*(Fig.2) How "mathematical" operator can describe various phenomena, rich in variety ??*

In the present quantum field theories including condensed matter physics, standard model, and string theory, they completely **rely** on "mathematical" ( **NOT** physical ) operators to describe various phenomena.

As you see this page, these creation and annihilation operators do **NOT** have the power to express various **dynamic** and complicated phenomena in the actual world.

Because, when a particle is created ( or annihilated ), these operators **cannot** designate various differernt **places** and **times** in each different dynamic particle.

So the **main** obstructions to the development of sicence are these very **abstract** oparators (and "spin ") !

*(Fig.3) Real variable world is NOT mathematical and simple operators !*

As you see various physical phenomena and **complicated** mechanism in our human bodies, actual particles such as electrons and protons are **moving around
**, interacting with each other at various **different** places and times.

For example, when a particle A is approching particle B, other another C is getting away from particle D ....

And various interactive "**timing**" and positional **relationship** determine various biological functions, which is **rich** in variety.

Unfortunately, the present mathematical operators such as "a^{†}" and "a" **cannot** describe these **variable** functions at all.

*(Eq.33)*

Also in Green's function (= propagator ), these **abstract** operators are used.

Eq.33 is Schrodinger equation of free particle.

*(Eq.34)*

" ± " means **anticommutation** and **commutation** relations of eigenfunctions (= operators ).

The eigenfunction in Schrodinger equation is

*(Eq.35)*

In Eq.35 "k" is wave number, and ħk is equal to a particle's momentum p.

ε means ( kinetic ) **energy** of a particle.

Substituting Eq.35 into Eq.34, we find "a" and "a^{†}" satisfy the following relation of

*(Eq.36)*

Green's function means the process in which a particle is created at ( t', x' ) and annihilated at ( t, x ) obeying the time order (= T ).

*(Eq.37)*

From Eq.35 and Eq.37, this Green function can be expressed using **Heaviside step function** of

*(Eq.38)*

This is used for time-ordered.

In the step function (= θ ) of Eq.38, when (t-t') is positive, θ becomes "1", and when t-t' is negative, θ becomes zero.

*(Eq.39)*

As shonw in this page, using complex integral, step function can be expressed as

*(Eq.40)*

In Eq.40, ε is infinitesimal ( ε → 0 ).

Using Eq.35, Eq.36, Eq.40, Green function of Eq.39 becomes

*(Eq.41)*

Here we do the replacement of

*(Eq.42)*

As a result, Eq.41 becomes

*(Eq.43)*

By the definition of delta function,

*(Eq.44)*

From Eq.35, Eq.43, and Eq.44, we get

*(Eq.45)*

This is Green function (= propagator ) in Schrodinger version.

2013/9/15 updated. Feel free to link to this site.