Top page (correct Bohr model )

Strange "spin" is NOT a real thing

Singlet Triplet don't mean "spin".

- Condensed matter physics = unreal quasiparticles.
- How unrealistic quasiparticles are made ?
- "Spin Peierls" is just mathematicak trick.
- Why "imaginary" time in phonon's Green function.

*(Fig.1) Many unreal quasiparticles ! = Condensed-matter physics.*

Most textbooks say the quantum mechanics is most successful theory, which can explain various latest technology.

But in fact, now in this 21th century, this **ancient** quantum mechanics clearly **obstructs** the development of science.

Researchers in these fields just aim at **illusory** particles, and **wasting** extremely much time.

As shown in Fig.1, the current condensed matter physics depends on many **imaginary** quasiparticles.

Researchers try to create **convenient** quasiparticles to fit some "math" models to experimental results.

These are just tricks, **NOT** real objects at all, as I say in this page.

To be precise, Bogoliubov particle (= γ ) in BCS theory is one of quasiparticle **fermions**.

And "phonon" is quasipaticle "**boson**". Of course, both are **unreal**.

Surprisingly, **many** unreal particles appear in this quantum mechanics. ( See Wiki. ) Very **disastrous** fact.

*(Fig.2) "Spin" and "Charge" are separated ??*

In this unreal solid state physics, "**Spin**" and "**Charge**" of one particle can be **separated**, they insist.

These are called "spinon" and "holon".

Of course, as shown on this site, these quasiparticles **don't** really exist.

But they try **NOT** to clarify what these **unreal** quasiparticles really are (= Shut up and calculate ! ).

As a result, only very **abstract** math symbols like Fig.2 are left in the latest quantum mechanics.

Furthermore, this site says they succeeded in **separating** electron into **three** components such as "Spin", "Charge" and "**Orbiton**" (= orbital motion ).

So the present condensed matter physics is very **far** from **real** physics, and just aiming at **fantasy**.

*(Fig.3) Fermions without "Spin" !? *

Surprisingly, the present condensed matter physics utilizes **spinless** fermions. ( See this site p.19 or this site . )

As you know, fermion **always** has spin 1/2.

There are NO spinless fermions in the current quantum mechanics.

So these theories are clearly **self-contradictory**.

Of course, as I explain in this page, these spinless fermions are just math ( NOT physics ) symbols, have **NO** reality.

The problem is that they try **NOT** to investigate what these spinless fermions really are (= Shut up and calculate !, again. )

So under the existing circumstances, the science **stops** at these unreal quasiparticles.

*(Fig.4) Interaction between Phonon and electron !?*

The present superconductivity including high temperature is basically based on quasiparticle **phonons** and Cooper pair.

But as you see Fig.4, these phonons and their interactions are just abstract math **operators**.

There are **NO** concrete physical images at all.

As you know, actual particles in this world are **NOT** math operators.

*(Fig.5) Phonon mechanism can be replaced by an integer multiple of de Broglie wavelength.*

The reason why they believe phonons and **Cooper** pair is **quantized** magnetic flux (= Φ = **h/2e** ).

( As far as I check, this is the **only** direct experimental result of this Cooper pair. )

As shown on this page, this phonon based Cooper pair uses vector potential A.

In fact, this Cooper pair phonon also relies on an integer multiple of **de Broglie** wave as "**phase**" .

( They should make clear what this "phase" **really** is, as soon as possible. )

On this page, we can prove this quantized magnetic flux means **real** de Broglie wave and **classical** Coulomb force.

The important difference between this de Broglie theory and phonon mechanism is "**reality**".

We **don't** need phonons in this flux quantum, because they are **unreal**.

*(Fig.6) "Imaginary" time is real ??*

In fact, all mechanisms involved in "phonon" rely on unrealistic **imaginary** time ( t = -iτ ).

Propagator (= Green's function ) is often used to calculate the change of the states with time.

Of course, all physical phenomena in this real world change with **real** ( NOT imaginary ) time.

So the current condensed matter physics is completely **out** of touch with reality.

*(Fig.7) "Imaginary" time is necessary for incorporating Boltzmann factor.*

At finite temperature, Boltzmann factor gives **probability** of some states with respect to their energy (= H ).

They try to incorporate this Boltzmann factor (= blue line ) into propagator, so they transform real time into imaginary !

This **artificial** method is for getting some discrete frequencies ( of course, "imaginary" frequency ).

These **discrete** frequencies based on Boltzmann factor are completely **different** from continuous frequencies of real time.

As you know, **all** phenomena in this real world are changing with **real** time.

So the calculated results based on this imaginary time are just **fictions**.

QCD also depends on imaginary time, so both these theories **lack** reality.

*(Fig.8) Time gap without computers made condensed matter physics "unrealistic" math.*

In 1920s when quantum mechanics was born, there were **NO** computers.

So the researchers at that time were **NOT** able to compute three-body helium atom using computers.

Of course, if three body helium couldn't be treated, it is much more **impossible** to compute more **complicated** condensed matter, which contains almost **infinite** atoms

But even in these serious situations, researchers **had to** continue researching and publishing papers for better **posts**.

So many kinds of unrealistic quasiparticles were artificially created as mathematical tools.

In high energy physics, unrealistic **10-dimensional** string theory is left as the **only** unified theory. This is disaster.

As you notice, these **fictitious** theories were needed to **fill** time gaps from 1920 to 1990s -.

*(Fig.9) Condensed matter physics = Ancient documents filled with math "hieroglyph".*

In about **1980**, the Apple II was produced as the first all-in-one computer.

But we needed to wait for more powerful computers until **1990s** to deal with **complicated** molecular interactions.

As I said, this extremely **long** ( 1920 - 1990 ) time **gap** caused many **unrealistic** theories.

Even after highly efficient computers appeared in 1990s, the conventional math ( NOT physical ) quasiparticles and **old** methods were left like **ancient** spells.

At first glance, you will find these old approaches become ancient **documents**, and **NOT** effective at all now in 21th century.

Because quasi- and virtual particles themselves are **unreal**.

*(Fig.10) How "fantasy" theories were made ?*

Quantum mechanics has **NO** reality.

But there were no other theories left under the conditions of NO computers in 1920s.

So even various ( top ) journals deal with these **fictitious** phenomena now in 21th century.

These theories are completely **unrealistic**, so they made **NO** progress at all ( ex. quantum computer ).

The serious problem is various mass **media** such as TV news, webnews, and newspapers are **promoting** these **fantasy** theories.

They are **deceiving** ordinary people, and giving **bad** influences to them.

They **must** make efforts to report experimental results as they are, **NOT** hiding their **defects**. ( This is the **true** role of mass media, I think. )

For example, Higgs news got quickly around the world last year.

But in spite of these vast news, ordinary people **don't** understand clear mechanism how Higgs gives mass to elementary particles.

Mass media try to report **ONLY** attractive parts, and **unrealistic** ones as much as possible.

*(Fig.11) They can deal with Only "Ratio" of Macro effects, NOT absolute values.*

Due to unrealistic spin and wavefunction, the current condensed matter physics **cannot** get into

All they can deal with are ONLY **macro** effects such as critical temperature, specific heat, magnetic susceptibility, and energy gap.

Furthermore, the present spin models are too abstract, so it is **impossible** to predict **absolute** values of them.

They just try to utilize many kinds of **artificial** quasiparticles and imaginary time to fit them with experimental results of "**Ratio**" ( NOT absolute values ! ).

As far as we are **bound** with incorrect models, we **cannot** get into **micro**worlds of actual particles' interactions.

*(Fig.12) Neutron scattering = spin wave or Phonon ??*

The direct experimental facts of unreal phenomena such as Phonons and Spin wave are ONLY **neutron** ( or light ) scattering.

They **don't** see these phonons and spin waves directly.

They just **estimate** their existences ONLY from the energy **change** of scattered neutrons.

( ex. Energy transition: neutron → phonon, phonon → neutron ).

Though phonons are just **quasiparticles**, which **don't** really exist.

*(Fig.13) There are No choices other than "Phonons" as atomic vibrations.*

Neutrons are scattered mainly from **nuclei** and magnetic moments.

Nuclear scattering is very complicated, because neutron is scattered by nuclear **internal** force.

So neutron scattering can be some sensor of nuclear (= atomic ) **vibration**.

But the present mathematical quantum mechanics **cannot** treat **many**-body effects, even if we use computers.

( Because, quantum mechanics was originally the theory, which is **independent** from computers. )

All they can depend on is very **abstract** math operator.

This is the main reason why they try to rely on **unreal** quasiparticles (= Phonons ) as atomic vibrations.

*(Fig.14) There are No choices other than "Spin" as magnetic moment.*

One of the two main scattering factors is **magnetic moment**.

Magnetic moment is very weak, but electron's orbital is spreading much **wider** than nuclei, so this effect is as big as nuclear scattering.

The important point is that the present condensed matter physics consider **ONLY** "Spin" as the origin of magnetic moments.

So if they observe some fluctuations of magnetic moments, they easily **jump** to the conclusion that these fluctuations are from "Spin".

They **neglect** any other large magnetic moment fluctuations such as **orbital** motions.

As shown on this page, these spin fluctuation theories are **filled** with mathematical tricks, and **lack** clear images.

*(Fig.15) Two fermions is equal to Four fermions ? Why ? Trick ? *

For spin Peierls and separation of spinon and holon, we need to use **artificial** mathematical tricks such as Fig.15.

Surprisingly, in Fig.15, they consider **two** fermions (= c^{†}c ) is equal to **four** fermions (= c^{†}cc^{†}c ).

So the equation of " **2 = 4** " holds, they insist.

Why does this strange situation occur ? ( See also this site . )

And you need to remember there are **NO** physical images from here.

Even the **latest** condensed matter physics is just **math** symbols, **NOT** physics.

*(Eq.1) *

In Eq.1, c^{†} and c mean creation and annihilation opearators of electrons.

"k" and "p" mean wavenumber and momentum. ( Here ħ is omitted, so, k = p. )

ε means energy.

*(Eq.2) *

These electron's operators satisfy anticommutation relation of Eq.2.

This relation is the origin of Pauli exclusion principle, they insist.

But as you see, these are just **math** rule **without** physical entity.

*(Eq.3) *

k_{F} and v_{F} denote Fermi wavenumber and velocity.

This Fermi energy (= E_{F} ) is most important concept, which is often used in the condensed matter physics.

But unfortunately, there are **NO** concrete physical images in these concepts.

*(Fig.16) Fermi energy is very vague concept. *

Orbitals become occupied in order from the lowest energy levels.

Each orbit contains two electrons due to Pauli exclusion principle.

The highest energy level of outer orbital is called "Fermi level ( energy )".

So these electrons in Fermi level **cannot** enter inner orbitals, which are already filled with other electrons.

*(Eq.4) Approximate expressions. *

Here we define like Eq.4.

Energy ε_{k} is a square of wavenumber "k"

So there are two patterns (= ± k ) in one energy ε.

In Eq.4 upper, this "k" is close to +k_{F}, and in Eq.4 lower, this "k" is close to -k_{F}.

Here, sign of ± is also expressed as R = +, and L = -.

*(Eq.5) Definitions of density ρ *

Here they define density operator ρ like Eq.5.

When the energy is close to Fermi energy, "k" inside ρ_{+} = ρ_{R} is close to +k_{F}.

( On the other hand, "k" inside ρ_{-} = ρ_{L} is close to -k_{F}. )

*(Eq.6) Strange commutation. *

As you see Eq.5, this density ρ consists of two fermions, so this behaves like "boson".

Even when momentum "q" is **different** from "-q", commutator of these ρ operators **doesn't** become zero. Why ?

This is strange.

*(Eq.7) *

In Eq.6, N_{q} is defined like Eq.7.

"q" is momentum, and "2π/L" is unit of wavenumber.

Substituting Eq.5 into Eq.6, and using anticommutating relation of Eq.2, we get Eq.8.

*(Eq.8) *

In Eq.8, "n" (= c^{†}c ) means number operator.

Suddenly, they change number operator into just "number" (= N_{q} ) in Eq.8. This is clearly a **trick**.

The important point is that they define the **lowest** wavenumber as k_{0}.

This is completely different from usual relativistic quantum field theory, which includes all wavenumbers (= momentums ) for Lorentz symmetry.

As a result, n_{k-q} contains **excessive** wavenumber between k_{0}-q and k_{0}, which gives the difference of N_{q}.

But as you notice, this method **doesn't** define **upper** wavenumber.

**Only** lower limit is considered, This is strange.

*(Eq.9) *

Here we consider commutation of Hamiltonian (= Eq.1 ) and ρ (= Eq.5 ).

For anticommutaion of "c", when c moves its neighbor, minus sign is added.

When c moves two operators, the sign is unchanged.

*(Eq.10) *

From Eq.9, we get Eq.10.

Eq.4 and ε = v_{F}k = v_{F}q are used.

*(Eq.11) *

Here, they define **new** Hamiltonian like Eq.11.

There are **NO** reasonable reasons why they suddenly introduce this Hamiltonian.

*(Eq.12) *

Of course, this is completely **artificial** definition.

Two fermions (= cc ) form one boson (= b ), so, they define **new** bosons (= b ) like Eq.13.

*(Eq.13) *

So they **like** to create new unreal bosons **only** for math tools.

This is what the present condensed-matter physics is.

Using new Hamiltonian of Eq.11 and relation of Eq.6, commutator of H_{B} and ρ becomes

*(Eq.14) *

The result of Eq.14 is **just** equal to Eq.10, though their Hamiltonian (= H_{0} and H_{B} ) are completely **different** forms.

But they suddenly accept new ( **wrong** ) equation of

*(Eq.15) Wrong equation.*

Commutation between Hamiltonian and ρ gives the same result, So these different Hamiltonian can be the same ?

As you see Eq.15, H_{0} contains only **two** operators (= c^{†}c ).

On the other hand, H_{B} contains as much as **four** operators (= c^{†}cc^{†}c ).

So the equation of H_{0} = H_{B} is completely **wrong**.

But this wrong equation is **indispensable** for getting ( unreal ) quasiparticles of Spinon, Holon, and spin Peierls.

*(Eq.16) Application of Wrong equation.*

Using the wrong equation of Eq.15, they got the quasiparticles of Spinon and Holon.

At first glance, the equation of Eq.16 is **unreasonable**.

H_{0} means **density**, on the other hand, H_{B} is **density × density**.

Here they choose this density × density form.

( In Eq.16, negative sign is omitted. )

*(Eq.17) *

In Eq.16, **σ** means "up" or "down" spins.

*(Eq.18) Separation of one particle into Spin and Charge ??*

In Eq.18, the sum of "up" and "down" spin particles means total **charge** (= ρ_{α} ).

And the difference between "up" and "down" spin particles means **spin** magnetic moments (= σ_{α} )

*(Eq.19) *

Substituting Eq.18 into Eq.19, we can get the Hamiltonian (= H_{B} ) of Eq.16.

This is the reason why they insist Eq.19 means the separation into spin (= spinon ) and charge (= holon ).

But as you see, these are just abstract math operators, and have **NO** physical properties at all.

Of course, it is **impossible** to separate one elementary particle into "spin" and "charge".

As shown on this site, they just observe some photoemission spectroscopy.

By linking these results to some **artificial** definitions, they just insist spin and charge are separated.

This is what the latest condensed matter physics is. (= **NOT** science at all ).

*(Eq.20)*

Here θ is defined like Eq.20 using new bosons of Eq.13.

Again, they introduced **other** new bosons (= Φ ) like Eq.21.

*(Eq.21) Again, new bosons ?*

You can find these Φ and Π satisfy usual commutation relation of Eq.22, using Eq.13, Eq.20 and Eq.21.

So they insist new **bosons** appear, again.

*(Eq.22) Commutator = new bosons ?*

where we use

*(Eq.23)*

And you find Hamiltonian (= H_{B} ) of Eq.11 can be expressed using Eq.21, like Eq.24.

*(Eq.24) Hamiltonian using new bosons ?*

As you see, the present condensed matter physics tries to take advantage of various **new bosons** as mathematical techniques.

There are **no** physical meanings here, they are going **away** from reality.

*(Eq.25) *

They define fermions ψ like Eq.25.

As you see, there are extremely many **artificial** definitions in the current quantum theory.

Commutator of b ( see Eq.13 and Eq.5 ) and ψ (= Eq.25) becomes

*(Eq.26) *

where they define

*(Eq.27) *

Redline of Eq.26 is anticommutation of Eq.2.

As a result, we have

*(Eq.28) *

*(Eq.29) Coherent states.*

When b and b_{†} are bosons and satisfy commutation relation, the equation of Eq.29 holds.

Here the formula of Eq.30 is used.

*(Eq.30) *

Comparing Eq.28 and Eq.29, they consider red and blue parts of them are just equal to each other.

This is very **forcible** interpretation, and just wrong.

*(Eq.31) *

Unfortunately, this equation is **NOT** true.

Finally, using both parts of Eq.29, they define

*(Eq.32) *

where θ_{R} is Eq.20.

Of course, equation of Eq.32 ( or Eq.31 ) is **wrong**.

Because

*(Eq.33) *

Bosons **cannot** be fermions, no matter how many bosons are combined.

So the current condensed matter physics is based on completely **wrong** assumptions **both** in mathematics and physics.

*(Eq.34) *

In the same way, we get Eq.34.

From Eq.32, Eq.34 and Eq.21, we get

*(Eq.35) *

As a result,

*(Eq.36) *

where we use

*(Eq.37) *

This is just mathematical trick.

They have already become out of touch with reality.

*(Fig.16) "Spin Peierls" transition really means "Spin" ?*

In spin-Peierls transition, **antiferromagnet** such as CuGeO3 becomes spin-**singlet** non magnet (= dimerization ) below some critical temperature (= Tsp ). See also this site

Of course, we **cannot** see unrealistic spin directly.

They just **imagine** these states **only** from magnetic **susceptibility** (= χ ) of Fig.16.

In antiferromagnet, susceptibility responding to external magnetic fields is much larger than singlet state, they insist.

In singlet state, two "up" and "down" spin atoms are tightly bound to each other ( of course, they **cannot** see these states directly. )

But they **NEVER** explain what these **strong** binding forces between spins really are.

As shown on this page, (anti) ferromagnetic energy is much **stronger** than spin magnetic dipole interaction.

*(Fig.17) Spinless fermions are necessary ?*

Surprisingly, the **spinless** fermions are necessary for spin-Peierls transition, they insist.

( See this site or this site )

Even in unrealistic supersymmetric theory, spinless fermions **don't** exist.

So from the beginning, these theories are **inconsistent**.

*(Eq.38) Nonlocal superposition of spinless fermions !?*

They define spin operators such as S^{+} and S^{-} like Eq.38.

In Eq.38, c_{i} means **spinless** fermion at the atomic position of "i".

Unfortunatelly, these spin model can treat ONLY **one** dimensional spin system for mathematical reason.

Even in simple one-dimension, they have to rely on some **appoximations**.

Surprisingly, these spin operators contains many **other** atomic spins.

Spins at **different** positions from 1 to i-1 influence spin at "i" (= nonlocal interactions ).

Unfortunately we **cannot** describe these nonlocal **spinless** fermions usin real objects.

*(Eq.39) *

They define S^{z} component like Eq.39.

As you see in Eq.38 and Eq.39, these fermions (= c ) have **NO** spins, so they can be used as the **common** fermions in **different** spin component operators.

And of course, these definitions are completely **artificial**.

*(Eq.40) *

Fermions obey anticommutation relation of Eq.2.

So two adjoining same operators like "cc" and "c^{†}c^{†}" become "**zero**".

As a result,

*(Eq.41) *

Using Eq.41, Eq.38 becomes

*(Eq.42) *

S_{x} and S_{y} components of spin can be expressed like Eq.43.

This transformation is called "Jordan -Wigner transformation"

*(Eq.43) *

*(Eq.44) XXZ one-dimensional spin model.*

XY model is so simple that we can solve it completely.

But XY model with no z component spin **cannot** describe real atomic states.

XXZ Heisenberg spin model is often used in spin Peierls transitions.

The problem is that this XXZ model **cannot** be solved, so we have to rely on some **rough** approximations and numerical calculation.

*(Eq.45) *

Substituting Eq.43 into Eq.44, we obtain Eq.45 ( x, y parts ).

And using the definition of Eq.42 and anticommutation relation, we have

*(Eq.46) *

and

*(Eq.47) *

Substituting Eq.39, Eq.46 and Eq.47 into Eq.45 and Eq.44, we have

*(Eq.48) *

In spin Peierls transition, uniformity of interatomic length needs to be broken.

So we have to adopt the following Hamiltonian.

*(Eq.49) *

In Eq.49, interatomic length at odd sites are different from that of even sites.

As you see, they just **intentionally** change the form of Hamiltonian to fit experimental results.

So these models are **NOT** natural results

*(Eq.50) *

Substituting Eq.50 into Eq.49, we have

*(Eq.51) *

where G= 2π × n and

*(Eq.52) *

In Eq.52, we need to depend on **artificial** transformations like Eq.15, Eq.24 and Eq.36.

As a result, Hamiltonian of Eq.51 becomes

*(Eq.53) *

where

*(Eq.54) *

See Eq.20.

And

*(Eq.55) *

These methods are also called "phase Hamiltonian".

Fermi wavenumber is defined as

*(Eq.56) *

The problem is that these forms are **inconsistent** with other spin wave theories.

So they artificially **change** the coefficient of Eq.55 into

*(Eq.57) *

where

*(Eq.58) *

This is one of **manipulations**.

Here we consider the change from π/2,

*(Eq.59) *

We treat Hamiltonian of Eq.53 as variational function like

*(Eq.60) *

Coefficients α and β as **variational** parameters.

So these are **free** parameters, we can adjust freely.

Fourier componets of θ and π are

*(Eq.61) *

Substituting Eq.61 into Eq.60,
and suddenly they define new **artificial** bosons (= "a", and "b" ) like

*(Eq.62) New boson's generation. ← mathematical trick, again.*

Of course, these bosons are **unreal** quasiparticles.

But they define **new** vacuum **based** on these artificial bosons like Eq.63,

*(Eq.63) *

Applying vacuum of Eq.63 to Hamiltonian of Eq.60,

*(Eq.64) *

The result of energy is

*(Eq.65) *

In antiferromagnet, Δ ( see Eq.44 ) is positive, and η ( see Eq.58 ) are

( These Δ and η are **free** parameters we can manipulate. )

*(Eq.66) *

From Eq.65, index of B ( considering also B inside γ ) is

*(Eq.67) *

From Eq.55, B is proportional to δ (= difference of coupling constants at odd and even sites ).

Supposing this coupling constant between spins are proportinal to some variables λ and **distortion** (= u ) of each atomic positions, the energy of Eq.65 can be expressed as

*(Eq.68) *

Here we define η = 1 (= artificial choice, again ).

*(Eq.69) *

When each distortion (= u ) of atomic positions is very small, the decrease of energy (= 4/3 power of "u" ) is larger than the increase of distortion energies (= a square of "u" ), they insist.

But we **don't** know the value of λ of Eq.68.

And as you see, they use various artificial definitions and quasiparticles.

Unfortunately, these theories are very **far** from reality.

*(Eq.70) Boltzmann factor is classical mechanics.*

It is known that various classical mechanical theories apply in actual phenomena, and they are **basis** of the current condensed matter physics.

For example, Borltzmann factor, Drude's conductivity and Curie-Weiss law..

Here we derive Boltzmann factor (= **exp(-βE)** ) using classical mechanics.

In Eq.70, E_{a} and E_{b} are energies of each A and B independent systems.

Probabilities of these states are P_{a} and P_{b}, respectively.

T is temperature.

*(Eq.71) *

In Eq.71, E and P(E) are the total energy and probability.

Differentiating P(E) of Eq.71 with respect to E_{a}, we have

*(Eq.72) *

We also express this differentiation like

*(Eq.73) *

Eq.72 is equal to Eq.73.

*(Eq.74) *

Dividing both sides of Eq.74 by "P" of Eq.71, we have

*(Eq.75) *

In Eq.75, left side consists only of the variable "E", and right side consists only of E_{a}.

Considering two variables are **independent** from each other, both sides need to be some **constant** (= -β ).

As a result,

*(Eq.76) *

This is a famous Boltzmann factor.

This is completely **classical** mechanics.

*(Eq.77) *

As shown on this page, Fermi statistics becomes like Eq.77.

In Eq.77, ε is energy, and μ is Fermi energy.

As far as I see various methods in condended matter physics, they **don't** use Fermi statistics of Eq.77 as it is.

They **always** change Eq.77 into very **rough** approximation.

So we have **NOT** yet prove Eq.77 is really correct.

*(Eq.78) *

Here we consider the extremely low temperature ( T → 0 ).

When the energy (= ε ) of the system is higher than Fermi energy (= μ ),

*(Eq.79) *

The probability f(E) of Eq.77 becomes **zero** in the limit of T → 0.

And when the energy is lower than Fermi energy,

*(Eq.80) *

f(E) becomes "**1**" in the limit of T → 0.

So they insist Fermi particles fill orbitals lower ( not upper ) than Fermi energy.

But this approximation is so **rough** that we **cannot** judge whether Eq.77 is really correct or not.

*(Eq.81) *

Eq.81 is Sommerfeld expansion.

δ( ) is Delta function.

Again, this is very **rough** approximation.

*(Eq.82) *

Seeing the acute transition from Eq.80 (= 1 ) into Eq.79 (= 0 ) at the point of Fermi energy, you can understand the delta function of Eq.82.

( At ε = E_{F}, this delta function becomes infinity. )

On the other hand, Planck constant (= quantization of de Broglie wave ) and Boltzmann factor had been proved to be correct.

*(Eq.83) Planck constant and Boltzmann factor are correct.*

*(Eq.84)*

"Phonon" is the most important concept in the current condensed matter physics.

Hamiltonian of this phonon becomes like Eq.84.

Last term of Eq.84 means the interaction between phonon (= b ) and electron (= c ), they insist.

As you see, these interactions are **too abstract** to be applied to actual complicated phenomena

*(Eq.85) Coupling constant.*

Eq.85 is coupling constant between phonon and electron.

Unfortunately, these parameters are just virtual, and **don't** express real phenomena.

*(Eq.86)*

The meaning of the phonon's interaction term (= last term of Eq.84 ) is like Eq.86.

In Eq.86 left, the electron with momentum of "k+q" gives the energy (= q ) to phonon (= b ), and becomes "k".

( "b_{q}" means phonon with "**q**" is **annihilated**. )

The maeaning of Eq.86 right is the **same** as Eq.86 left.

"b_{-q}^{†}" means phonon with the momentum of "**-q**" is **created**.

*(Eq.87) Imaginary time is indispensable.*

In fact, the current condensed matter physics completely depends on virtual "**imaginary**" time for phonon's transition.

So these theories **lack** reality from the beginning.

Here we explain how this imaginary time (= - i τ ) is used in phonon's Green function.

Usual form of operator in Heisenberg (= interaction ) picture is

*(Eq.88)*

Here they suddenly change real time (= t ) into imaginary time (= τ ) like

*(Eq.89)*

Substituting Eq.89 into Eq.88, we have

*(Eq.90)*

Green function is propagator, which shows how the states will change with time.

In this **thermal** Green function, they see ONLY the direction of imaginary time.

( Though we want to know how the states change with **real** time. )

*(Eq.91)*

In Eq.91, θ( τ - τ' ) is step function.

( When τ - τ' > 0, this θ is "1", when τ - τ' < 0, this θ is "0". )

Two operators are in time order (= Tr, right is before, left is after. )

*(Eq.92)*

Eq.92 picks up only the last term of Eq.91.

In fermions, commutating gives negative sign. ( ± : + is boson, - is fermion. )

At finite temperature, the probability is given by Boltzmann factor (= exp(-βH) ) × Green function, as shown in Eq.92.

*(Eq.93)*

When we consider some Hamiltonian (= H ), eigenstates on left and right sides are the same.

If we use some matrix, this results correspond to **trace** of that matrix.

And the order of these matrices can be circulated like Eq.93 (= trace ).

Using Eq.93, Eq.92 becomes

*(Eq.94)*

In Eq.94, exponential functions containing H can be commuted.

Three operators on the left side are transferred to the right.

And adding two exponential functions (= exp(-βH) exp(βH) = "1" ) to Eq.94 from left side, we have

*(Eq.95)*

Like Eq.95, imaginary time **τ-τ'** is changed into **βħ + τ-τ'**, after incorporating Boltzmann factor into them.

The important point is that, we need to adopt artificial condition of Eq.96.

*(Eq.96)*

Under the condition of Eq.96, Green funtion become the same periodically at the interval of **imaginary** time βħ, they insist.

But as you see, we want to know how states change with "real" ( **NOT** imaginary ) time.

So these operations are just mathematical trick with **NO** reality.

*(Eq.97)*

Eq.97 is Fourier series of this Green function.

Considering periodicity with time, these **frequencies** (= ω ) become discrete like

*(Eq.98) This is just for math trick.*

In Fermion's Green function, negative sign is added, when two operators are exchanged.

So these frequencies become odd number × π in fermions

There are **NO** physical images in these tricks.

This is what the latest condensed matter physics really is.

2014/1/11 updated. Feel free to link to this site.