Top page (correct Bohr model including helium. )

Supersymmetry is a waste of money.

- Chirality -- left or right handed.
- Lorentz transformation of spinor.
- What is Majorana particle ?
- Supersymmetric transformation is artificial.

*(Eq.1) *

On this page, we use the notations of this site.

Eq.1 is the chiral expression of γ matrices.

Only on this page, ( **1, -1, -1, -1** ) version of metric tensor is used.

And σ is Pauli matrices, which are

*(Eq.2) *

From Eq.1, we have

*(Eq.3) *

Here we introduced 4 × 1 Dirac spinor Ψ, as follows,

*(Eq.4) *

Dirac equation is

*(Eq.5) *

where

*(Eq.6) *

Here we use natural unit, so

*(Eq.7) *

Multiplying Eq.7 by γ^{0} from left side,

*(Eq.8) *

where

*(Eq.9) *

"E" means energy, which is the time derivative .

And the momentum operator (= p ) is

*(Eq.10) *

Using Eq.10 and Eq.4, Eq.9 becomes

*(Eq.11) *

From Eq.11, we obtain two equations,

*(Eq.12) *

where

*(Eq.13) *

When fermion's mass is zero ( m = 0 ), Eq.12 becomes

*(Eq.14) *

As you see Eq.14, upper 2 component spinor ψ has **plus** helicity (= σp ).

This means the directions of spin (= σ ) and momentum (= p ) are the same. (= **right**-handed ).

On the other hand, the lower 2 component spinor (= χ ) has **minus** helicity.

So this is **left**-handed spinor.

*(Eq.15) *

If we apply γ^{5} matrix on spinor, upper and lower spinors give different signs.

*(Eq.16) *

To make Dirac equation Lorentz invariant, they needed to **define** Lorentz transformation of spinor.

They introduced the following S operator.

*(Eq.17) *

In Eq.17, ω is antisymmetric **constant**.

And σ^{μν} is commutator of two γ matrices.
( This σ is NOT Pauli matrix. )

*(Eq.18) *

So we have

*(Eq.19) *

Using Eq.18 and Eq.19, infinitesimal Lorentz transformation of Eq.17 becomes

*(Eq.20) *

In Eq.20, we express ω using different constants ε and η

Applying Eq.20 Lorentz operator to Dirac spinor of Eq.15,

*(Eq.21) *

From Eq.20 and Eq.21, we have

*(Eq.22) *

From Eq.20, the inverse matrix of Eq.20 is

*(Eq.23) *

Because

*(Eq.24) *

From Eq.1 and Eq.2, we have

*(Eq.25) *

From Eq.17, Eq.20, Eq.23, Eq.25 we obtain the following relation,

*(Eq.26) *

where we use

*(Eq.27) *

As a result,

*(Eq.28) *

So, Dirac Lagrangian is **invariant** under Lorentz transformation (= S ), as follows,

*(Eq.29) *

Like Eq.27, when we define

*(Eq.30) *

In Eq.30, Pauli matrices σ is Hermitian matix.

So the comlex conjugate transpose of σ remains the same ( σ^{†} = σ ).

From Eq.22 and Eq.30, we have

*(Eq.31) *

Lagrangian of Eq.29 can be expressed as

*(Eq.32) *

and

*(Eq.33) *

So, we can prove Eq.32 is scalar Lagrangian.

Here we use the notations of

*(Eq.34) *

From Eq.30 and Eq.31, we have

*(Eq.35) *

From Eq.2, each Pauli matrix satisfies

*(Eq.36) *

Because only σ_{2} consists of imaginary number.

From Eq.35 and Eq.36, we get

*(Eq.37) *

Eq.37 means this wavefunction transforms like ψ of Eq.31, as follows,

*(Eq.38) *

Like Eq.33,

*(Eq.39) *

Using Eq.38, Eq.39 becomes

*(Eq.40) *

In Eq.40, i* = -i, and σ_{2}* =- σ_{2} is used.

And

*(Eq.41) *

From Eq.41,

*(Eq.42) *

Here we define

*(Eq.43) *

From Eq.42 and Eq.43,

*(Eq.44) *

*(Eq.45) *

From Eq.44 and Eq.45,

*(Eq.46) *

Using Eq.44, Eq.46 becomes

*(Eq.47) *

E.47 is Lorentz invariant scalar.

χ and ξ are fermions, so anticommutator of Eq.47 isn't zero.

Here we introduce new operator of ε^{ab},

From Eq.43 and Eq.44, we have

*(Eq.48) *

So,

*(Eq.49) *

Eq.48 is equal to

*(Eq.50) *

So,

*(Eq.51) *

Using Eq.49, we can express

*(Eq.52) *

Under the external electromagnetic field, Dirac equation contains vector ( scalar ) potentials A, like

*(Eq.53) *

The complex conjugate (= * ) of Eq.53 is

*(Eq.54) *

Using Eq.1 and Eq.3, we get the relation of

*(Eq.55) *

Because,

*(Eq.56) *

Multiplying Eq.54 by ( iγ^{2} ) from left side,

*(Eq.57) *

The first term of Eq.57 can be expressed as

*(Eq.58) *

where we insert

*(Eq.59) *

Using Eq.55 and Eq.58, Eq.57 becomes

*(Eq.60) *

Comparing Eq.53 and Eq.60, you find the charge "e" of Eq.60 becomes the opposite from Eq.53.

(= **Charge conjugate** of ψ becomes iγ^{2}ψ*. Just **artificial** rule. )

When we define

*(Eq.61) *

We get

*(Eq.62) *

This means, when the form of the spinor is Eq.61, the charge conjugate of wavefunction is **equal** to the original one.

The charge conjugate is antiparticle, so in this case, the particle is the same as its antiparticle.

*(Eq.63) Majorana particle ?*

They call this state Majorana particle.

It is said that neutrino and photino are Majorana fermions.

But unfortunately, this is just **abstract** math operator, **lacks** physical images.

*(Eq.64) Supersymmetry Lagrangian ?*

Eq.64 is the simplist supersymmetry Lagrangian (= Wess-Zumino model ).

φ is boson, and χ is fermion ( here we consider simple Majorana fermion ).

And "F" is "unreal" **auxiliary** field (= boson ), which is indispensable for supersymmetry **algebra**.

These auxiliary fields were **artificially** introduced, which **proves** supersymmetry itself is just math ( **NOT** physical ) concept.

*(Eq.12) *

From the right Dirac equation of Eq.12, we have

*(Eq.65) *

this is

*(Eq.66) *

where we use

*(Eq.34) *

As shown on this site (p.33), units of total Lagrangian (= L ), boson (= φ ), fermion (= χ ), and F boson become

*(Eq.67) *

here we use the natural unit of

*(Eq.68) *

They define supersymmetric **transformation** of boson φ as

*(Eq.69) *

Of course, boson is NOT fermion.

So **two** fermions (= ξ, χ ) are necessary for expressing one boson (= φ ).

From Eq.67 and Eq.69, the unit of this **strange** fermion ξ is

*(Eq.70) *

So, this ξ is completely different from the ordinary fermion χ of Eq.67.

Depending on these **unreal** fermions and bosons proves supersymmetry itself is **NOT** truth.

*(Eq.71) *

And they define supersymmetry transformation of fermion χ (= Eq.71 ) and boson F (= Eq.72 ).

Of course, these forms were **artificially** introduced, and **NOT** natural ones.

*(Eq.72) *

Transformation of boson part in Lagrangian (= firs term of Eq.64 ) becomes

*(Eq.73) *

In Eq.73, Eq.69 is inserted.

From Eq.64 and Eq.71, fermion's transformation becomes

*(Eq.74) *

Here we use the antisymmetric property of Pauli matrices.

And integration by parts gives negative sign.

*(Eq.75) *

Using Eq.36,

*(Eq.76) *

As a result, Eq.74 becomes

*(Eq.77) *

Again, the integration by parts gives negative sign, as follows,

*(Eq.78) *

Using Eq.72, auxiliary parts FF transforms like

*(Eq.79) *

From Eq.73, Eq.77 and Eq.79, the change of total Lagrangian becomes **zero**,

*(Eq.80) *

Eq.80 means Lagrangian of Eq.64 is **invariant** under supersymmetric transformations of Eq.69, Eq.71 and Eq.72.

This is called "**supersymmetry**".

You find physicists try to define artificial rules, and make Lagrangian invariant **by force** under supersymmetry transformation.

So SUSY does **NOT** mean the law of Nature at all, it exists **only** inside math equation.

Basically, Lagrangian of MSSM (= minimal supersymmetric standard model ) becomes

*(Eq.81) *

The last term of Eq.81 is potential energy.

And "D" is covariant derivative.

*(Eq.82) *

In Eq.82, A is vector potential, or gauge bosons.

Due to the **strict** limitation of SUSY and Lorentz invariance, the form of W must be

*(Eq.83) *

Furthermore we can add the Lagrangian of vector potential ( ex. photon ),

*(Eq.84) *

λ of Eq.84 is "photino". Just math ( **NOT** physical ) symbols.

And antisymmetric tensor of Eq.84 is

*(Eq.85) *

The interactive Hamiltonian between **Higgs** (= H ) and other particles are

*(Eq.86) *

Eq.86 gives "mass" to particles including Higgs itself.

As supersymmetry must be **broken**, the mass Lagrangian of SUSY particles is

*(Eq.87) *

Due to this SUSY breaking, as many as **124** free parameters appear even in MSSM.

So supersymmetry **cannot** predict anyhing at all.

MSSM define R parity.

Of course, this R parity is just **artificial** rule, **NOT** a natural result.

*(Eq.88) *

Ordinary particles always have R = +1, and **SUSY** particles have **R = -1**.

For example, in ordinary quark, B (= baryon ) number is 1/3, L (= lepton ) number is zero,
and spin S is 1/2.

*(Eq.89) *

The left of Eq.89 is R parity of ordinary quark, and the right of Eq.89 is squark.

Spin S of squark is **zero**, so R parity becomes -1.

The current **useless** quantum field theory considers **ONLY** invariance of Lagrangian L ( or action S ) under some **transformation**.

The variation of action S is expressed as

*(Eq.90) *

Here we use Euler-Lagrange equation of

*(Eq.91) *

Eq.90 becomes

*(Eq.92) *

Integral of total derivative becomes zero, as shown in Eq.90.

This means, when Euler-Lagrange equation holds, action S is **invariant** under some transformation.

If we define artificial current J as

*(Eq.93) *

Eq.92 can be expressed as

*(Eq.94) *

Eq.94 means, if we define **artificial** charge Q as

*(Eq.95) *

This total charge is conserved.

SUSY theorists considers this **unreal** charge Q as most imporntant concept. This is strange.

Generally, arbitrary transformation can be expressed using this charge Q, as follows,

*(Eq.96) *

Here we use Lagrangian of

*(Eq.64)*

The canonical momentum π of boson field φ is

*(Eq.97) *

Here we define commutation relation between this φ and its canonical momentum (= π ).

*(Eq.98) *

From Eq.95 and Eq.64, the charge Q becomes

*(Eq.99) *

Using Eq.98 and Eq.99, we have

*(Eq.100) *

We can prove Eq.96.

*(Eq.101) *

When there are several kinds of transformations (= Q ), the anticommutator of these Q give the **central charge** Z of Eq.101.

To cancel this abstract central charge, superstring theory needs unreal **10 dimensions**, as shown on this page.

So there are **NO** physical reality in these SUSY and string theory.

They are just math symbols.

Basicaly, rotational operator can be expressed as

*(Eq.102) *

"J" is angular momentum, and θ is angle.

In case of spinor rotation, J becomes Pauli matrices (= σ/2 ),

*(Eq.103) *

Charge "Q" of SUSY is a spinor, so it changes like

*(Eq.104) *

If we define

*(Eq.105) *

we have

*(Eq.106) *

and

*(Eq.107) *

As a result,

*(Eq.108) *

Eq.108 means the charge Q can **decrease** particle's spin 1/2.

But SUSY theorists NEVER try to say what this charge Q ( = spin stropper !? ) really is.

*(Eq.109) *

In case of complex comjugate of Q,

*(Eq.110) *

Spin 1/2 is **increased**.

*(Eq.111) *

If this charge Q is applied to graviton with spin "2", it changes into **gravitino** with spin 3/2, they insist.

Unfortunately, there are **NO** other concrete explanations about spin here.

*(Eq.112) *

When there are 4 kinds of supersymmetric transformation, this is called "N=4" supersymmetry.

In this case, the number of different Q is "4".

2014/5/1 updated. Feel free to link to this site.