• GENERAL RELATIVITY & COSMOLOGY for Undergraduates
• Lattice gauge theory
• Stringy black holes, by Jan Pieter van der Schaar

Least action principle

The laws of physics can be deduced from the principle of least action, which says that the action S is minimized.
The lagrangian L is the action per unit of time.
The lagrangian density Λ is the lagrangian per unit of volume.
The lagrangian density multiplied by a 4-dimensional volume of time-space gives an action.
The fact that the action is minimized is expressed by δ S = 0, which is equivalent to d/dt (∂ L/∂ (dq/dt)) - ∂ L / ∂ q = 0.

Proof :
δ S = δ ∫_t1^t2 L dt = ∫_t1^t2 δ L dt = ∫_t1^t2 ((∂ L/∂ q) δ q + (∂ L/∂ (dq/dt)) δ(dq/dt)) dt
∫_t1^t2 (∂ L/∂ (dq/dt)) δ(dq/dt) dt = ∫_t1^t2 (∂ L/∂ (dq/dt)) d(δ q)/dt dt
= ∫_t1^t2 d/dt ((∂ L/∂ (dq/dt)) δ q) dt - ∫_t1^t2 d/dt (∂ L/∂ (dq/dt)) δ q dt
because d/dt ((∂ L/∂ (dq/dt)) δ q) dt = d/dt (∂ L/∂ (dq/dt)) δ q dt + (∂ L/∂ (dq/dt)) ((∂(δ q))/dt) dt
From δ S = ∫_t1^t2 ((∂ L/∂ q) δ q + (∂ L/∂ (dq/dt)) δ(dq/dt)) dt
and ∫_t1^t2 (∂ L/∂ (dq/dt)) δ(dq/dt) = ∫_t1^t2 d/dt ((∂ L/∂ (dq/dt)) δ q) dt - ∫_t1^t2 d/dt (∂ L/∂ (dq/dt)) δ q dt follows :
δ S = [(∂ L/∂ (dq/dt)) δ q]_t1^t2 + ∫_t1^t2 (∂ L/∂ q - d/dt (∂ L/∂ (dq/dt))) δ q dt
[(∂ L/∂ (dq/dt)) δ q]_t1^t2 = 0 because δ q(t1) = δ q(t2)
δ S = 0 for all variation of δ q
Then d/dt (∂ L/∂ (dq/dt)) - ∂ L/∂ q = 0.

Other proof :
δ L = ∂ L/∂ q δ q + ∂ L/∂ (dq/dt) δ (dq/dt) + ∂ L/∂ t δ t
δ t = 0 => δ L = ∂ L/∂ q δ q + ∂ L/∂(dq/dt) d/dt δ q
δ S = ∫ ∂ L/∂ q δ q dt + ∫ ∂ L/∂(dq/dt) d/dt δ q dt
∫_t1^t2 ∂ L/∂(dq/dt) d δ q = [∂ L/∂(dq/dt) δ q]_t1^t2 - ∫_t1^t2 d/dt ∂ L/∂ q δ q dt
δ S = ∫ (∂ L/∂ q - d/dt ∂ L/∂(dq/dt)) δ q dt = 0
∂ L / ∂ q - d/dt ∂ L/∂(dq/dt) = 0

If we define the momentum p = ∂ L / ∂ (dq/dt), this equation can be written : dp/dt = ∂ L / ∂ q.

See Degenerate Lagrangians and Legendre transformations - http://viswiz.gmd.de/~nikitin/course/node2.html :

"Lagrangian mechanics works in n-dimensional configuration space, which includes all parameters, defining a static state of mechanical system (coordinates of particles, orientation of rigid body etc). A point x in this space draws a curve x(t) in evolution. For such curves a functional A[x(t)], called action, is introduced. Only those curves, on which the action reaches an extremum, correspond to real evolution.

Usually consideration is restricted to functionals of the form

A[x(t)] = ∫ dt L(x,dx/dt)

with Lagrangian L dependent only on coordinates and velocities. The condition of extremum for the action leads to Lagrange-Euler equations

d/dt (∂ L/∂(dx/dt)) - ∂ L/∂ x = 0

This is (normally) a system second order differential equations, with solutions uniquely defined by initial coordinates and velocities x(0), dx/dt (0).

In Hamiltonian mechanics the state of the system is described by a point (x,p) in 2n-dimensional phase space. The dynamics is defined by a function H(x,p), called Hamiltonian, via equations:

dx/dt = ∂ H / ∂ p , dp/dt = - ∂ H / ∂ x

Look here for alternative formulation of the same theory.

Transition from Lagrangian to Hamiltonian mechanics is performed by Legendre transformation. It defines the momenta and Hamiltonian as:

p(x,dx/dt) = ∂ L / ∂ (dx/dt) , H(x,p) = p dx/dt - L .

The Hamiltonian depends on coordinates and momenta, so one should express the velocities via momenta, inverting the definitions of momenta: dx/dt = dx/dt(x,p) and substitute the result into Hamiltonian. "

See also Poisson brackets and symplectic form - http://viswiz.gmd.de/~nikitin/course/node3.html

"A binary operation { , } , called Poisson brackets, is introduced for functions on the phase space:

  {A, B} = ∂ A/∂ x ∂ B/∂ p - ∂ A/∂ p ∂ B/∂ x

In terms of Poisson brackets the Hamiltonian equations can be rewritten as

dx/dt = {x, H} , dp/dt = {p, H}.

Exercise: prove, that for any function A(t,x,p) the following identity (Liouville equation) is valid:

dA/dt = ∂ A/∂ t + {A, H}. "

To determine the laws of physics for some phenomenon, one can express the action, the lagrangian or the lagrangian density corresponding to this phenomenon and apply this equation to it.

Example : particle moving in a gravitational field in non-relativistic approximation (v<<c) :

• q = x (position)
• dq/dt = dx/dt = v (speed)
• m is the mass of the particle
• g is the acceleration of the gravitational field

Remark : mimimizing the action means maximizing the potential energy. It means that the trajectory is higher that the straightforward line, it is REPELLED by the mass. Generally one consider that the particles are ATTRACTED by masses. This paradox is because generally we consider the initial position and the initial speed as fixed, and the trajectory is lower than the straightforward line corresponding to this initial position and speed, but here the initial and final positions are fixed, and in this case the trajectory is higher than the straight line between these positions.

Conservation on energy deduced from the least action principle :
dL/dt = ∑_\a ∂ L/∂ q_a dq_a/dt + ∂ L/∂(dq_a/dt) d² q_a/dt²
∂ L/∂ q_a = d/dt ∂ L/∂(dq_a/dt)
dL/dt = ∑_a d/dt ∂ L/∂(dq_a/dt) dq_a/dt + ∂_L/∂(dq_a/dt) d² q_a/dt² = ∑_a d/dt (∂ L/∂(dq_a/dt) dq_a/dt
d/dt (∑_a ∂ L/∂(dq_a/dt) dq_a/dt - L) = 0
E = ∑_a ∂ L/∂(dq_a/dt) dq_a/dt - L
dE/dt = 0

Path integral :
x(t1) -> x(t2)
e^{i/hbar S}

Particules et lois de la physique, par Richard Feynman et Steven Weinberg
Conférence de Steven Weinberg : A la recherche des lois ultimes de la physique

Dans un univers mythique où les seules particules sont l'électron et le photon, la densité de lagrangien est :
L = - psi bar (gamma^mu d/dx^mu + m) psi - 1/4 (dA_nu/dx^mu - da_mu/dx^nu)^2 + ieA_mu psi bar gamma^mu psi - mu (dA_nu/dx^mu - dA_mu / dx_nu) psi bar sigma^mu^nu psi - G psi bar psi psi bar psi + ...

avec :

• psi = champ associé à l'électron
• m = masse de l'électron
• gamma et sigma sont des matrices
• A = champ du photon

Théorie des cordes :
métrique g_alpha_beta(sigma) de la surface décrivant l'évolution de la corde dans le temps
action I[x,g] = 1/2 intégrale (g(sigma))^1/2 g^alpha^beta(sigma) dx^mu(sigma)/dsigma^alpha dx^nu(sigma)/dsigma^beta d^2 sigma.

Special relativity

(1  0  0  0)

(0 -1  0  0)

(0  0 -1  0)

(0  0  0 -1)

The scalar product of two vectors u^μ = (u⁰ u¹ u² u³) = (u^t u^x u^y u^z) and v^ν is g_μν u^μ v^ν = ∑_μ=0³ ∑_ν=0³ g_μν u^μ v^ν.

Gauge fields

Example of the electromagnetic field

The electromagnetic field is a gauge field associated with U(1) which is simply the circle.
It can also be viewed as gravitation in the fifth dimension of a 5 dimensional space obtained by multiplying the 4 dimensional space-time by the circle U(1) which is compactified to the Planck length. This is Kaluza-Klein theory.

The phase is a point on the circle U(1).
The potential A_m = (A_t=V A_x A_y A_z) is the gradient of the phase. In the general case, the potential is A^a_m where a is one of the compactified dimensions of the phase space, but in the case of the electromagnetic field, the circle U(1) has only 1 dimension, so this indice a is useless.
The field F_mn is d_mA_n-d_nA_m where d is the partial derivation. It means that F_mn is the dephasage of an infinitesimal loop in the plane mn.

    S(t+dt,x) <--- -Ax(t+dt,x+dx/2) --- R(t+dt,x+dx)

    |                                   ^

    |                                   |  

    -At(t+dt/2,x)                       +At(t+dt/2,x+dx)

    |                                   |    

 t  V                                   |

 ^  P(t,x) ------- +Ax(t,x+dx/2) -----> Q(t,x+dx)

 |

 +--->x

• Let us start at point P(t,x) with phase φ₀.
• At point Q(t,x+dx) the phase is φ₁ = φ₀ + A_x(t,x+dx/2)dx.
• At point R(t+dt,x+dx) the phase is φ₂ = φ₀ + A_x(t,x+dx/2)dx + A_t(t+dt/2,x+dx)dt.
• At point S(t+dt,x) the phase is φ₃ = φ₀ + A_x(t,x+dx/2)dx + A_t(t+dt/2,x+dx)dt - Ax(t+dt,x+dx/2)dx.
• Back to point P(t,x), the phase is φ₄ = φ₀ + A_x(t,x+dx/2)dx + A_t(t+dt/2,x+dx)dt - Ax(t+dt,x+dx/2)dx - A_t(t+dt/2,x).

In the general case the field is :
F_mn^a = d_mA_n^a - d_nA_m^a + f_bc^aA_m^bA_n^c
where m and n are dimensions of space-time (t,x,y,z), a, b, c are dimensions of phase space (compactified dimensions) and f_bc^a are the structure constants.

The field satisfies d_mF_np + d_pF_mn + d_nF_pm = 0 or e^mnpq d_nF_pq = 0.

The potential A_m is determined by the current density of the sources J_m = ρ v_m (where ρ is the charge density and v the speed) by the relation : [] A_m = μ₀J_m with [] = 1/c d²/dt² - d²/dx² - d²/dy² - d²/dz².

The field is also directly determined from the sources by
d_nF^mn = -µ₀J^m

A particle moving in an electromagnetic field is submitted to a force. f = e F v

( γ/c f.v )     (0    Ex/c Ey/c Ez/c) (γ c )

( γ fx    ) = e (Ex/c 0    Bz   -By ) (γ vx) 

( γ fy    )     (Ey/c -Bz   0    Bx ) (γ vy)

( γ fz    )     (Ez/c By   -Bx  0   ) (γ vz)

With a metric (+ - - -), F_mn =

(0     Ex/c Ey/c Ez/c)

(-Ex/c 0    -Bz  By  )

(-Ey/c Bz   0    -Bx )

(-Ez/c -By  Bx   0   )

F^mn =

(0    -Ex/c -Ey/c -Ez/c)

(Ex/c 0     -Bz   By   )

(Ey/c Bz    0     -Bx  )

(Ez/c -By   Bx    0    )

Kaluza Klein theory

Notations :

• g_mn is the metric of the 4 dimensional space-time
• Φ is the dilaton which is the metric of U(1)
• g'_mn is the metric of the 5 dimensional space-time

g'_mn = ( g_mn - Φ A_m A_n   - Φ A_m )

        ( - Φ A_n            - Φ     )



g'^mn = ( g^mn   - A^m        )

        ( - A^n  - 1/Φ + A^2)

Generalization to any gauge field

• Gravity : no gauge field, it is the curvation of 4 dimensional space-time
• Electromagnetism : U(1)
• Weak interaction : SU(2)
• Strong interaction : SU(3)
• Unification EM-weak-strong : SU(5)

Conventions : mu' = { μ, m } where

• μ are the 4 dimensions of space-time t, x, y, z
• m are the internal coordinates of compactified dimensions
• mu' are all the coordinates of the generalized space-time

E'^a'_mu' = ( e^a_μ          0     )

            ( A_μ^k E^n_k    E^n_m )



e'_a'^mu' = ( e_a^μ    -e_a^ρ A_ρ^m )

            ( 0         E^n_m            )

g'_mu'nu' = ( g_m_n + G_m_n A_μ^m A_ν^n   A_μ^m G_m_n )

            ( A_ν^n G_m_n                  G_mn         )



g'^mu'nu' = ( g^μ^ν   -A^μ^n               )

            ( -A^ν^m   G^mn + A^ρ^m A_ρ_n)



or more symetrically :



g'^mu'nu' = ( g^μ^ν            -A_ν^n g^μ^ν              )

            ( -A_μ^m g^μ^ν    G^mn + g^μ^ν A_μ^m A_ν^n ) 

A charged particle moves in the fifth dimension with a speed proportional to its charge divided by its mass.
Two particles which have the same charge move in the same direction in the fifth dimension. According to special relativity, their mass increase because of this speed, so the attractive force should be greater, but from the point of view of these particles, they do not move. The increasement of the attractive force is compensed by a repulsive force : the electrostatic force.

According to the gauge field theory, the force fⁱ = m aⁱ which acts on a particle of mass m, charge q and speed v^j in an electromagnetic field F^ij is : fⁱ = q v^j Fⁱ_j.
According to Kaluza Klein theory, the particle follows geodesics of 5-dimensional space-time, which means that the acceleration aⁱ = d² x / dt² = - Γⁱ_jk v^j v^k, then fⁱ = m aⁱ = - m Γⁱ_jk v^j v^k.
The affine connection Γ is :
Γⁱ_jk = 1/2 g^il (∂_j g_lk + ∂_k g_lj - ∂_l g_jk)
F_ij appears in Γⁱ_ju and Γⁱ_uj where u is the compactified Kaluza Klein dimension.

Lagrangian of a particle in an electromagnetic field

Action S = -mc ∫ ds

ds² = c²dt² - (dx²+dy²+dz²)

ds = c 2√(1-v²/c²)dt

L = 1/2 mv² + e A.v - eV
where

• L is the lagrangian
• m is the mass of the particle
• v is its speed
• e is its charge
• A is the vector potential
• V is the scalar potential

S = - ∑ mc ∫ ds - ∑ e ∫ A_mdx^m + 1/2μ₀ ∫_t1^t₂dt ∫(E²/c²-B²)d τ
= - ∑ mc ∫ ds - ∑ e ∫ A_mdx^m + 1/2μ₀ ∫_t1^t₂dt ∫(-1/2 F_mnF^mn)d τ

L = -1/4 F_mnF^mn + A_mJ^m

L = √(-g) (1/4 R - 1/2 g_ij(φ) ∂_μφⁱ ∂^μφ^j -1/4 m_IJ(φ) F^μνI F_μν^J - 1/8 ε^μνρσ a_IJ(phi) F_μν^I F_ρσ^J)

With metrics (- + + +) :
L = -1/4 F_μν F^μν + J_μ A^μ + L_matter
L = -1/4 F_μν F^μν + ψ bar (γ^μ (∂_μ + i e A_μ) + m) ψ
J^μ = ∂ L / ∂ A_μ = - i e ψ bar γ^μ ψ
V(t) = + i e ∫ d³ x (ψ bar(x,t) γ^μ ψ(x,t)) a_μ(x,t) + V_coul(t)

See also Lagrangian Theory of Fields and Particles

Lagrangian Theory of Fields and Particles

--> postscript only (10 pages)

Abstract

This is a discussion of how to do particle and field Lagrangians. The particle's action principle seeks to maximise the proper time in a trajectory between two events in spacetime. The field's action principle seeks to minimise the integral over whatever Lorentz scalar can be constructed from the field, given its symmetry, and with the proviso that the resulting Hamiltonian be positive definite. The interaction arises from simple coupling, to the particle itself in the case of a scalar field, and through the particle's four velocity in the case of a vector field. We will see that the vector field coupled to the particle's motion in this way must give rise to an antisymmetric force field, which by symmetry of interaction forms the basis for the construction of Lorentz scalars. A very fundamental result is that like charges attract for scalar field interactions, while they repel for the vector field. The vector field, of course, is the basis for electromagnetism. An often repeated error concerning the particle's equation of motion in a scalar field is cleared up -- since the four velocity is a unit vector, the force must be projected orthogonal to the world line in order to have consistent dynamics. It is interesting to note that most of the properties of electrodynamics, including the repulsion or attratction of charges, are necessary consequences of linearity, masslessness, and the fact that it is a vector field.

Tips and tricks that you learn from this include how to decide what the signs should be for the free particle and field parts of the Lagrangian, how to manipulate the variation of the proper time, why the sign of the interaction part is arbitrary, how to convert back and forth between a Lagrangian and a Lagrangian field density and likewise between particles and ensemble densities, how to make to correspondence to the more common nonrelativistic model, and how to get all the factors of the speed of light, c, properly when you don't assume natural units with c equal to unity. These are all things I had trouble with and so I wanted to write them up together in a coherent whole. The result should be useful to any physicist or physics watcher interested in the basis for modern theory.

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• Lectures Page

See http://www.uwm.edu/~norbury/gr/node37.html

Classical Field Theory

In both special and general relativity we always seek covariant equations in which space and time are given equal status. The Euler-Lagrange equations (4.6) are clearly not covariant because special emphasis is placed on time via the dq_i/dt and d/dt(∂ L/∂ (dq_i/dt)) terms.

Let us replace the q_i by a field φ ≡ φ (x) where x ≡ (t, x) . The generalized coordinate q has been replaced by the field variable φ and the discrete index i has been replaced by a continuously varying index x. In the next section we shall show how to derive the Euler-Lagrange equations from the action defined as  

S ≡ ∫ L dt

(148)

which again is clearly not covariant. A covariant form of the action would involve a Lagrangian density L via  

S ≡ ∫ L d4 x = ∫ L d3 x dt

(149)

with L ≡ ∫ L d³ x . The term - ∂ L/∂ q_i in equation (4.6) gets replaced by the covariant term - ∂ L/∂ φ (x) . Any time derivative d/dt should be replaced with ∂_μ ≡ ∂ / ∂ x^μ which contains space as well as time derivatives. Thus one can guess that the covariant generalization of the point particle Euler-Lagrange equations (4.6) is  

∂μ ∂ L / ∂ (∂ μ φ ) - ∂ L/∂ φ = 0

(150)

which is the covariant Euler-Lagrange equation for scalar fields. This will be derived rigorously in the next section.

In analogy with the canonical momentum in equation (4.5) we define the covariant momentum density  

Π μ ≡ ∂ L/∂ (∂μ φ )

(151)

so that the Euler-Lagrange equations become  

∂μ Πμ = ∂ L/∂ φ

(152)

The canonical momentum is defined as  

Π ≡ Π0 = ∂ L/∂ dφ/dt

(153)

The energy momentum tensor is (analagous to (4.8))  

Tμν ≡ Πμ ∂ν φ - gμν L

(154)

with the Hamiltonian density

• Classical Klein-Gordon Field

Classical Klein-Gordon Field

The energy momentum tensor is
Therefore the Hamiltonian density is H≡ T₀₀=(dφ/dt)²-1/2(∂_α φ ∂^α φ -m² φ ²) which becomes [31]
where we have relied upon the results of Section 3.4.1.

See http://www-th.phys.rug.nl/~halbers/science/verslag/node5.html

Next: The Hamiltonian formalism Up: Symmetries in field theory Previous: Symmetries in field theory

The Lagrangian formalism

The dynamics of a set of classical fields φⁱ(x^μ) on some d-dimensional manifold M can be obtained from a Lagrangian density L(φⁱ,∂_μφⁱ) through the principle of stationary action

The Hamiltonian formalism

There is an alternative approach to classical field theory in which one considers a Hamiltonian H which can be obtained from the Lagrangian by

Noether's theorem

Suppose now that the action is invariant under some internal global symmetry group G which we shall assume to be a Lie group. Then there exists a set of generators T_a satisfying the Lie algebra of G

In the above we considered internal symmetries but Noether's theorem applies equally well to spacetime symmetries such as translations and rotations. The conserved charges are then the energy-momentum and the angular momentum of the system under consideration.

The charge algebra

So far we have been discussing classical field theory. In making the transition to quantum field theory it is convenient to use the Hamiltonian rather than the Lagrangian formalism. One can then postulate the fields and the momenta to be operators on a suitable Hilbert space and define canonical equal-time commutation relations between them

In quantum field theory Noether's theorem remains valid but in addition the charges Q_a satisfy the same Lie algebra as the generators T_a

   
Local symmetries

Up to now we have only discussed global symmetries where the infinitesimal parameters did not depend on the spacetime co-ordinates. In making the transition to local symmetries one makes the parameters co-ordinate dependent. In doing so, the group-algebra remains the same as do the transformation rules for the fields. However, the derivatives ∂_μ no longer commute with the group transformations

In making a Lagrangian with a global symmetry group invariant under the local version of these symmetries, one replaces all ordinary derivatives ∂_μ by covariant derivatives D_μ and then adds the invariant quantitiy F^a_μνF_a^μν to this new Lagrangian.

Local symmetries are for historical reasons usually called gauge-symmetries and physical models exhibiting such symmetries are called gauge-theories. In particular, if the gauge-group is a non-Abelian internal symmetry group, then such models go under the name of Yang-Mills theories.

Quantum theory of fields

Π_l(x,t) = δ L[Ψ(t),.Ψ(t)]/δ.Ψ^l(x,t)

equations of motion :
.Π_l(x,t) = δ L[Ψ(t),.Ψ(t)]/δ Ψ^l(x,t)

Action I[Ψ] = ∫_-∞^∞ dt L[Ψ(t),.Ψ(t)] = ∫ d⁴x L(Ψ(x),∂Ψ(x)/∂ x^μ)

∂/∂ x^μ ∂ L/∂ (∂ Ψ^l/∂ x^μ) = ∂ L/∂Ψ^l

H = ∑_l ∫ d³x Π_l(x,t) .Ψ^l(x,t) - L(Ψ(t),.Ψ(t))

L = -1/2 ∂_μφ ∂^μφ - m²/2 φ² - H(φ)

With a metrics (- + + +) : (  - m²) φ = H φ

Π = ∂ L / ∂ .ψ = φ

H = ∫ d³x (Π .φ - L) = ∫ d³x[1/2 Π² + 1/2 (∇ψ)² + 1/2 m² φ² + H(φ)]

L = -1/4 F^μν F_μν - Ψ bar (γ^μ (∂_μ + i e A_μ) Ψ

J^μ = ∂ L / ∂ A_μ = - i e Ψ bar γ^μ Ψ

Non abelian gauge fields

Gauge transformation :
ψ_n(x) -> e^ie_nΛ(x)ψ_n(x) for arbitrary Λ.
A_μ(x) -> A_μ(x) + ∂_μΛ(x)
Gauge covariant derivative : ∂_μψ_μ(x) - i e_n A_μ(x) ψ(x)

δψ_l(x) = i ε^α(x)(t_α)_l^mψ_m(x) where t_α are matrices.
[t_α,t_β] = i C^γ_αβ t_γ
C^γ_αβ = - C^γ_βα
0 = [[t_α,t_β],t_γ] + [[t_γ,t_α],t_β] + [[t_β,t_γ],t_α]
0 = C^δ_αβ C^ε_δγ + C^δ_γα C^ε_δβ + C^δ_βγ C^ε_δα
(t^A_α)^β_γ = -i C^β_γα
[t^A_α,t^A_β] = i C^γ_αβ t^A_γ

Example : Yang-Mills : ψ = (ψ_p, ψ_n) wave functions of proton and neutron

t1 = 1/2 (0 1)   t2 = 1/2 (0 -i)   t3 = 1/2 (1 0)

         (1 0)            (i  0)            (0 1)

t1A = (0  0  0)   t2A = (0  0  i)   t3A = (0 -i  0)

      (0  0 -i)        (0  0  0)        (i  0  0)

      (0  i  0)        (-i 0  0)        (0  0  0)

δ(∂_Mψ_l(x)) = iε^α(x)(t_α)_l^m(∂_μψ_m(x)) + i(∂_με^α(x))(t_α)_l^mψ_m(x)

δ A^β_μ = ∂_με^β + i ε^α(t^A_α)^β_γ A^γ_μ
δ A^β_μ = ∂_με^β + C^β_γα ε^α A^γ_μ

D_μψ(x)_l = ∂_μψ_l(x) - i A^β_μ(x)(t_β)_l^mψ_m(x)
= iε^α(t_α)_l^m∂_μψ_m - i C^β_γα ε^α A^γ_μ(t_β)_l^mψ_m + A^γ_μ(t_γ)_l^m(t_α)_mⁿψ_n

δ(D_μψ)_l = i ε^α(t_α)_l^m (D_μψ)_m

F^γ_νμ = ∂_ν A^γ_μ - ∂_μ A^γ_ν + C^γ_αβ A^α_ν A^β_μ
δ F^β_νμ = iε^α(t^A_α)^β_γ F^γ_νμ = ε^α C^β_γα F^γ_νμ

L(ψ,D_μψ,D_νμψ,...F^α_μν,D_ρ F^α_μν,...)
∂ L/∂ψ_l i(t_α)_l^mψ_m + ∂ L/∂(D_μψ_l) i(t_α)_l^m)(D_μψ_m) + ∂ L/∂(D_nu D_μψ_l) i(t_α)_l^m (D_ν D_μψ_m) + ... + ∂ L/∂ F^β_μν C^β_γα F^γ_νμ + ∂ L/∂ D_ρ F^β_μν C^β_γα D_ρ F^γ_νμ + ... = 0

L_A = -1/2 g_αβ F^α_μν F^βμν
g₁₁ = g₂₂ = -g₃₃ = -2

L = -1/4 _αμν F_α^μν + L_m(ψ,D_muψ)
Equations of motion : ∂_μ(∂ L/∂(∂_μ A_αν)) = - ∂_μ F_α^μν = ∂ L/∂ A_alpha_ν = - F_γ^νμ C_γαβ A_βμ - i ∂ L_μ/∂(D_νψ) t_αψ

Complete current j :
∂_μ F_α^μν = -j_α^ν
j_α^ν = -F_γ^νμ C_γαβ - i ∂ L_μ / ∂ (D_νψ) t_α ψ
∂_ν j_α^ν = 0
D_λ F_α^μν = ∂_λ F_α^μν - i (t_β^A)_αγ A_βλ F_γ^μν
= ∂_λ - C_αγβ A_βλ F_γ^μν
=> D_μ F_α^μν = -J_α^ν
J_α^ν = -i ∂ L_μ / ∂ (D_νψ) t_αψ
[D_ν,D_μ] F_α^ρσ = - i (t^A_γ)_αβ F_γνμ F_β^ρσ = - C_γαβ F_γνμ F_β^ρσ
D_ν J_αν = 0
D_μ F_ανλ + D_ν F_αλν + D_λ F_αμν = 0

D_μψ = ∂_μψ - i t_α A_αμ ψ
Constraints : Π_α0 = ∂ L / ∂ (∂₀ A⁰_α) = 0 and -∂_μ ∂ L / ∂ (∂_μ A_α0) + ∂ L/∂ A_α0 = ∂_μ F_α^μ0 + F_γ^μ0 C_γαβ A_βμ + J_α⁰
= ∂_k Π_α^k + Π_γ^k C_γαβ A_βk + J_α⁰ = 0
Π_α^k = ∂ L / ∂ (∂₀ A_αk) = F_α^k0

General relativity

When a vector v^μ is transported from x + dx to x, it's coordinates are augmented by Γ^μ_νρ v^ν dx^ρ where Γ is the connection or the Christoffel symbol.

Covariant derivative :
D_mA^l = d_mA^l + Γ^l_mnAⁿ
D_mA_l = d_mA^l - Γⁿ_mlA_n

D_μ g_νρ = 0 => Γ^l_mn = 1/2 g^ls ( d_mg_sn + d_ng_sm - d_sg_mn)

Curvature R^b_amn = d_mΓ^b_an - d_nΓ^b_am + Γ^l_an Γ^b_lm - Γ^l_am Γ^b_ln
R_am = R^l_alm
R = g^amR_am

Action : S_g for gravitation, S_e for matter and electromagnetism.
δ S_g = -1/(2χ c) ∫ (R_ij - 1/2 g_ij R) √(-g) δ g^ij dΩ
S_g = -1/(2 χ c) ∫ 2√(-g) R d Omega with χ = 2 π G / c⁴

With matter and electromagnetic field (Λ_e) :
action S_e = 1/c ∫ √(-g) Λ_e dΩ
T_ij is defined by : 1/2 T_ij √(-g) = ∂(√(-g) Λ_e) / ∂ g^ik - ∂_l ( ∂ (√(-g) Λ_e) / ∂ (∂_l g^ik) )
Then δ S_e = 1/c ∫ 1/2 T_ik √(-g) δ g^ik dΩ.
Then δ (S_g + S_e) = 0 becomes :
-1/(2χ c) ∫ (R_ij - 1/2 g_ij R - χ T_ij) √(-g) δ g^ij dΩ = 0
R_mn - 1/2 g_mnR = χ T_mn
R_ij = χ (T_ij - 1/2 g_ij T)
Λ_(em) = -1/(4μ₀) F_ik F^ik
T_(em) = 1/μ₀ (- F_ri F_sⁱ + 1/4 g_rs F_ik F^ik)

See also :

ESPACE-TEMPS FRACTAL ET MICROPHYSIQUE

Ainsi la relativité restreinte mène à la contrainte qu'aucune vitese ne peut excéder une certaine vitesse universellec, qu'on peut ensuite identifier à la vitesse de n'importe quelle particule de masse nulle dans le vide, en particuler la lumière.¹⁹ On rappelle que l'espace-temps minkowskien est caractérisé par l'invariant:

ds² = c²dt² - (dx² + dy² + dz²) ,

dans n'importe quelle transformation de coordonnées inertielles.

Poursuivons avec l'évolution des idées en relativité. Ce fut l'une des contributions les plus importantes de Mach que d'insister sur la relativité de tous les mouvements, par seulement des mouvements inertiels. En se fondant sur les principes de covariance générale et d'équivalence, Einstein construisit la théorie de la relativité générale, dont les équations peuvent être vues comme des contraintes sur les courbures de l'espace-temps qui sont physiquement possibles. Les équations d'Einstein

R_μν - (1/2) R g_μν - Λ g_μν = χ T_μν (2.1)

sont les plus générales des équations les plus simples qui soient covariantes sous toutes les transformations de système de coordonnées continues et (au moins deux fois) différentiables.

Les lecteurs qui souhaiteraient un exposé complet de cette théorie sont invités à consulter des livres tels ceux de Misner, Thorne et Wheeler²² ou Weinberg.⁵⁵ Rappelons simplement ici brièvement que, dans ces équations, les g_μν sont des potentiels de métrique tensoriels qui généralisent le potentiel gravitationnel newtonien scalaire. L'invariant de la relativité générale s'écrit, avec la convention d'Einstein de sommation des indices identiques présents en haut et en bas

ds² = g_μν dx^μ dx^ν, (μ,ν= 0 à 3).

En relativité générale, l'existence de la courbure de l'espace-temps implique que les variations des quantités physiques telles que les vecteurs ou les tenseurs dans des transformations infinitésimales de coordonnées dépendent de l'espace-temps lui-même. Ce fait s'exprime par la dérivation covariante

D_μA^ν= ∂_μA^ν+ Γ^ν_ρμ A^ρ

qui généralise la dérivation partielle usuelle. Dans cette expression, les effets de l'espace-temps (c'est-à-dire de la gravitation) sont décrits par les symboles de Christoffel

Γr_mn = (1/2) g^ρλ ( ∂_ν g_λμ + ∂ _μ g_λν - ∂_l g_μν ) ,

qui jouent le rôle du champ gravitationnel. Les dérivées covariantes ne commutent pas, si bien que leur commutateur conduit à l'apparition d'un tenseur à quatre indices, le tenseur de Riemann R^λ_μνρ :

(D_μ D_ν - D_νD_μ ) A_ρ = R^λ_rnm A_λ .

La contraction du tenseur de Riemann donne le tenseur de Ricci R_μν = g^λρR_λμρν:

R_μν = ∂_ρΓ^r_μν - ∂_νΓ^ρ_μρ + Γr_mn Γ^λ_ρλ - Γ^ρr_μλ Γ^λ_νρ ,

tandis que la quantité R = g^μν R_μν est la courbure scalaire. Les équations d'Einstein énoncent que le tenseur énergie-impulsion T_μν est égal, à la constante χ = 8πG/c⁴ près, au tenseur géométrique d'Einstein donné par le premier membre de l'équation (2.1), dans laquelle Λ est la constante cosmologique. Le tenseur d'Einstein et le tenseur énergie-impulsion se conservent au sens covariant. Le principe d'Einstein de l'équivalence de la gravitation et de l'inertie s'exprime par le fait qu'on peut toujours trouver un système de coordonnées dans lequel la métrique est localement minkowskienne, et que dans un tel système les équations du mouvement d'une particule libre sont celles du mouvement inertiel, Du_μ = 0, où u_μ est la quadri-vitesse de la particule. Ecrite dans un système de coordonnées quelconque, cette équation devient l'équation des géodésiques

(d²x^μ/ds²) + Γm_nr (dx^ν/ds) (dx^ρ/ds) = 0 .

Dirac equation

(i d/ - m) Ψ(x) = 0
L_QED = Ψ bar (i D/ - m) Ψ = - Ψ bar (i <-D/ + m) Ψ
with D_m = d_m + i e A_m
and (for any Z) Z/ = γ^mZ_m
and γ are the matrices :

γ 0 = (1   0)

      (0  -1)



γ k = (0       σ k)

      (-σ k    0  )



σ 1 = (0 1)

      (1 0)



σ 2 = (0 -i)

      (i  0)



σ 3 = (1  0)

      (0 -1)

L_QCD = i ∑_i=1^N_F q bar_i(x) (d_m - ig_s ∑_a=1⁸1/2l^aA_m^a(x)) γ^mq_i(x) - 1/4 ∑_a=1⁸F_mn^a(x) F^mna(x) - ∑_i=1^N_F q bar_i(x) M_iq_i(x)
where

• q_i is the quark field
• A_m^a is the gluon field
• i is the flavor of the quark
• a is the flavor of the gluon
• N_F is the number of flavors
• M_i are the quark masses
• l^a are the Gell-Mann color matrices
• γ^m are the Dirac matrices

See also :

• The Dirac Equation - http://tjd-online.simplenet.com/Dirac/Dirac.html
• Scalable implementation of spectral methods for the Dirac equation, by J. C. Wells - http://gopher.ccs.ornl.gov/staff/wells/index.html/sc98/sc98.html
• Dirac equation and Potential - http://physik.kfunigraz.ac.at/~jom/bag/node4.html
• Charles Bloom's page : The Dirac Equation : Intro to RQM - http://lonestar.texas.net/~cbloom/physics/dirac.html
• The Generalized Dirac Equation - http://kaluza.physik.uni-konstanz.de/2MS/laemmerz/la/node2.html

With a metrics (+ - - -)

the Klein-Gordon equation is : ([] + m^2) Ψ(x) = 0

with [] = d/dt^2 - d/dx^2 - d/dy^2 - d/dz^2



Let γ^μ (i=0..3) be 4 matrices :



γ^0 = ONE 0

      0 -ONE 



γ^k = 0       σ^k

      -σ^k 0



ONE = 1 0 

      0 1



σ^1 = 0 1

      1 0



σ^2 = 0 -i

      i  0



σ^3 = 1 0

      0 -1



Let A/ be (sum over μ = 0,1,2,3 of ) γ^μ A_μ



The Dirac equations are :

	(i d/ - m) Ψ(x) = 0

and :

	i (d_μ Ψbar) γ^μ + m Ψbar = Ψbar (i <-d/ + m) = 0

With an electromagnetic field :

	i d Ψ/dt = (α (p - e A) + e PHI + β m) Ψ

	i d/dt Ψ(r,t) = (-i α ∇ + β + e αha . A(r,t) - e A^0(r,t)) Ψ(r,t)

	(α p + β (m + U(x)) Ψ(x,t) = i d/dt Ψ(x,t)



The lagrangian is :

	L = Ψbar (i d/ - m) Ψ

	= - Ψbar (i <-d/ + m) Ψ

        = 1/2 (i Ψbar <-d/ Ψ - i Ψbar d/ Ψ) - m Ψbar Ψ

	= 1/2 (i Ψbar γ^μ d_μ Ψ - i d_μ Ψbar γ^μ Ψ) - m Ψbar Ψ



	L_DEM = Ψbar (i D/ - m) Ψ = Ψbar (i d/ - e A/ - m) Ψ

	with D_μ = d_μ + i e A_μ



	L_QED = -1/4 F_μ_ν F^μ^ν + Ψbar (i D/ - m) Ψ 

i hbar d psi/dt = H psi
H = -i hbar c alpha . nabla + a_4 mc^2
-hbar^2 d^2psi/dt^2 = H^2 psi

i hbar ∂ψ/∂ t = Hψ
H = -i hbar cα∇ + a₄ m c²
-hbar² ∂²ψ/∂ t² = H²ψ = - hbar² c² α_i α_j ∂²ψ/(∂ x_i ∂ x_j) - i hbar m c³ (α_i α₄ + α₄ α_i) ∂ψ/∂ x_i + m² c⁴ α₄²ψ
α_i α_j + α_j α_i = 2 δ_ij 1
α_i α₄ + α₄ α_i = 0
α₄² = 1

α1 = 0 0 0 1

     0 0 1 0

     0 1 0 0

     1 0 0 0 



α2 = 0  0  0 -i   

     0  0  i  0

     0 -i  0  0

     i  0  0  0



α3 = 0  0  1  0

     0  0  0 -1

     1  0  0  0

     0 -1  0  0



α4 = 1  0  0  0

     0  1  0  0

     0  0 -1  0

     0  0  0 -1

σ = 0 0 1 0 α

    0 0 0 1

    1 0 0 0

    0 1 0 0

String theory

A string is described by a parametrized surface X^m(τ,σ)
The action of a free bosonic string moving in curved space is :
S = 1/2α' ∫ d τ d σ 2√(-h) h^ab d_aX^m d_bXⁿ g_mn(X)
where h^ab is the metric of the wordsheet (τ,σ).

See also Introduction to String Theory

Rishons

See Mathias Home - Explaining the surplus of matter with the Rishon-model

The Rishon-Model

The Rishon model was developed by Haim Harari, Professor for high energy physics at the Weizmann Institute in Rehovot/ Israel and Nathan Seiberg.
The main idea is that quarks and leptons of the first generation {u (up-quark), d (down-quark) and e^- (electron), n_e (electron-neutrino) } each consist of three particles - the rishons or their anti-particles.
The model assumes two new particles (rishons) and their anti-matter counterparts (anti-rishons). They were named after the state of the earth in the beginning of the creation (1. Mose), Tohu and Vavohu (desert and empty). The symbols are T and V.

Rishon	Charge
T	+1/3
-T	-1/3
V	0
-V	0

Rishons and anti-rishons can be combined in triples. There is only one restriction: Rishons and anti-rishons may not be combined together in one particle (quark or lepton).

Rishon-combination	Particle	Name	Charge
T T T	e+	positron	+1
T T V	u	up-quark	+2/3
T V V	-d	anti-down-quark	+1/3
V V V	ne	electron-neutrino	0
-V -V -V	-νe	anti-electron-neutrino	0
-V -V -T	d	down-quark	-1/3
-V -T -T	-u	anti-up-quark	-2/3
-T -T -T	e-	electron	-1

There are eight possible combinations, corresponding to the quarks and leptons of the first generation and their anti-particles.
The model considers the color of the particles. There are three color states for each rishon:

For example TTT (e⁺) is a colorless lepton, containing rishons of one color each resulting in white.
Quarks are colored combinations, the colors cannot compensate to white.

Particles with charges -2/3, -1/3, 1/3 and 2/3 are colored.
Particles with the charges -1, 0, +1 are colorless.

The colors describe a very strong force that does not allow any colored particle to stay alone. Stable particles must be white, the particles they consist of have to compensate each other to white.

The rishon-theory offers a very interesting answer to the question why proton and electron have exactly the same absolute charge (1.60217733*10^-19C). There is only ane particle that carries charge in the rishon-theory: the T rishon. Electron and proton consist of this rishon and its anti-rishon.

Further conclusions from the Rishon-Model

What are the particles, a hydrogen atom consists of?

particles	description
e- + p	electron + proton
e- + u u d	electron + quarks (proton)
-T -T -T T T V T T V -V -V -T	rishons (electron) + rishons (proton)

There are 4 T rishons and 4 -T , 2 V rishons and 2 -V .
These rishons can be combined to one electron, one positron and one d/-d-pair as well. And that would be pure energy!

the same can be done with the anti-hydrogen atom:

particles	description
e+ + -p	positron + anti-proton
e+ + -u -u -d	positron + quarks (anti-proton)
T T T -T -T -V -T -T -V V V T	rishons (positron) + rishons (anti-proton)

From the level of the rishons, matter and anti-matter consist of the same particels. They can theoretically be converted into each other or into energy.
This is a very important aspect of the following description of how there can be more matter than anti-matter in the universe.

Next: Four-dimensional Actions and Dual Up: String Effective Actions Previous: Kaluza Klein Theory

   
Reduction of the Low Energy String Effective Action

The method discussed in the previous section can be generalised to include more dimensions and more complicated fields. We can use it to reduce our ten-dimensional string effective action sea10 to four dimensions. But let's consider the general case of an effective action in D-dimensions that we want to reduce to (D-d) dimensions. So in the simplest case we assume that space-time is of the form , where M has D-d dimensions and K has d dimensions and all fields are independent of the coordinates y^m of K.

This the simplest way to compactify the extra dimensions and of course there are more interesting ways to do this compactification which have more realistic features, but they are also far more difficult. For the moment we will stick to the case in which the fields are taken to be independent of the extra K coordinates.

We will use the following notation: Local coordinates of M are ( ), internal coordinates of K are y^m, ( ). The total space has signature ( ) and fields in this D-dimensional space are denoted with a hat, as well as their coordinates (, , etc.), ( ). Quantities without a hat are then the (D-d)-dimensional ones.

We begin with the first terms of the low energy string effective action seff, the Einstein term coupled to the Dilaton:

(3.18)

It is very useful to do this reduction with the aid of so called vielbeins (vierbeins or tetrads in four dimensions, vielbeins in any other dimension). With the use of these vielbeins we make the connection between curved coordinate systems and local Lorentz (flat) coordinates. See appendix for a short introduction of these vielbeins.

On the same transformation grounds that we saw in the previous section we can determine the reduction of these vielbeins. Let again the greek indices denote the curved indices, ( ) where m labels the internal coordinates. And the hatted roman indices will denote the Lorentz indices in D-dimensions, ( ). We assume that the internal space is flat, so there is no difference between the internal 'curved' indices and the internal Lorentz indices. The vielbeins reduce in the following way [5]:

(3.19)

(3.20)

(3.21)

We can put these expressions into the Einstein part of our effective action r1 and after a tedious calculation one finds

S =
			(3.22)

We also have to reduce the other part of our string effective action involving the anti-symmetric field tensor

(3.23)

We can derive that the following reduction of the D-dimensional anti-symmetric field tensor gives us the correct transformation properties in (D-d) dimensions

(3.24)

Note that is a gauge tensor field: . This means that the (D-d)-dimensional fields will have the following transformation properties:

	=
	=

We can use the vielbeins to convert the curved indices of to Lorentz indices and then use the reduced vielbeins ( and Eⁿ_m) to convert them back to (D-d)-dimensional curved indices:

	=
	=		(3.25)

	=		(3.26)
	=
	=		(3.27)
	=
	=		(3.28)

Due to the dimensional reduction there arise extra terms in the definition in the field strength tensor . These are the so called Abelian Chern-Simons terms.

We can put all these terms back into the action. The total reduced effective action, consisting of the Einstein part as well as the part with the anti-symmetric field tensor, can be written in a very symmetric form [5]. The claim is that there is a O(d,d) global symmetry that keeps this action invariant. (see Appendix on the symmetry group O(d,d).)

First we have to introduce the scalar matrix M,

(3.29)

(3.30)

(3.31)

(3.32)

	=
	=
	=
	=

(3.33)

So the reduction of a Low energy String Effective Action in D dimensions to D-d dimensions results allways in a action redsea which has an O(d,d) symmetry. This dual symmetry is called T-Duality, where the 'T' stands for Target-space. In the case of the string effective action of the bosonic sector of the Heterotic string sea10 we have to reduce from D=10 to four dimensions. This means d=6 and the reduced action will be invariant under O(6,6) transformations. This also means there will be 12 vector fields and M will be a matrix with 36 independent scalars.

As we mentioned before, we expect that compactification on more complicated manifolds K will give more realistic features in four dimensions. In fact there are very many ways to perform this compactification and it is not clear yet which of them should be the correct way.