Last dream of Einstein

Published by Rahul Aggrawal on

Last dream of Einstein (Gauge Theory of Gravity)

The last dream of Einstein was to unify the four fundamental forces. It has   not been possible yet because scientists could not unify gravity with other forces. Other three forces follow rules of quantum mechanics except for gravity.

Gravity is the only force in which no quantum rules are applicable. The theory in which the other three forces are unified known as the grand unified theory (GUT) or QFT and QCD. Quantum Field theory and Quantum Chromodynamics.

There is no quantum theory of gravity because gravity is the theory of geometry. To quantize gravity we have to quantize spacetime geometry or geometrize the quantum mechanics.

Some mathematicians are working in this area. They are known as the string theorists. They are using the higher dimensional concept to geometrize quantum field theory.

One of the important candidates to unify gravity with other forces is Gauge theory of gravity. There is a similarity between the gravitational field and non- abelian gauge field known as Yang Mills field.

To study gravity as a gauge field we have to understand group theory and the fibre bundle. Yang mills potential is a connection form in the fibre bundle.


Meaning of Gauge Theory:

Let us discuss first the meaning of gauge theory. Gauge Theory is a theory of symmetry and certain invariance principle. One of the important gauge theory is the theory of electricity and magnetism given by Maxwell. In the Maxwell theory, electric and magnetic fields are observables which depend upon scalar and vector potentials.

If we are adding certain parameter known as a gauge in the potentials then E and B will remain same. So Gauge means some type of transformations so lagrangian remains invariant. To study simple gauge transformations in electromagnetism David J. Griffith is a good book. 

In the Electromagnetism gauge field (scalar and vector potential) can vary and field strength tensor remains the same. This field strength tensor is sometimes known as curvature of gauge fields. It is the exterior derivative of the Maxwell field or gauge field(A).

Actually, the gauge transformations are a large set of variational symmetries. Physically, the gauge transformation symmetry has no physical content. Maxwell electromagnetic field theory is an abelian gauge theory. It means the gauge field or vector potential belongs to the representation of the Lorentz group.

Maxwell’s equations are invariant under gauge transformations. The formulation of the equation is in covariant tensor form. The field strength tensor remains unchanged under the gauge transformation of the four vector potential (known as a gauge potential).  

Such a theory which remains invariant under local gauge transformations known as gauge theory and to quantize this theory we consider gauge potential or four-vector potential as an operator or dynamic variable.
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Non Abelian Gauge Theory:

Non Abelian Gauge Theory: Weak interactions and QCD are examples of non-abelian gauge theory. Gauge Theory of weak interactions is very fascinating and related to spontaneous symmetry breaking and Higss mechanism that we will discuss later on. QCD or quantum chromodynamics is the gauge theory of strong interactions. Interaction particle or exchange particles known as gauge boson here.

Maxwell theory is much simple than this because it is based on abelian gauge group U(1). But QCD theory involve more complicated symmetries. Here gauge groups are non commutative so known as non-abelian gauge group. Broadly these theory known as Yang-Mills theories.

Yang Mills Theory: 

Yang Mills Theory: Gauge theory arises when we are trying to promote global symmetry to a local symmetry. As in quantum electro dynamics, Dirac Lagrangian density for free fermion has global symmetry. The internal symmetry in this casearise because the fermionic field is complex field variable and lagrangian density is Hermitian. Phase ofthe field can be arbitrary. If we want to promote local symmetry than neither lagrangian density nor the corresponding action remains invariant under local phase transformations.

To solve this problem we introduce a new field variable known as gauge field. Physically it describe the minimal coupling of charged fermions with electromagnetic field. Now lagrangian becomes invariant under local gauge transformations. But Gauge field is non dynamical in this theory.

To give it dynamicity in a gauge invariant manner we define field strength tensor and introduce in QED lagrangian density. And in this way local gauge invariance described.

Now we are trying to generalise this idea to free Dirac theory. Here also lagrangian density remains invariant under global phase transformations. A group associated here are Lie group. Lie group are a symmetric non-commutative group.

In the free field Dirac Theory, Dirac field belongs to a non trivial representation of some internal symmetry group G.

This global phase transformation we can express in terms of infinitesimally small real global parameter with generators of the group. Generators of the group should be Hermitian. The generators satisfy the Lie algebra of the Lie group G.

Because generators of the symmetry group do not commute, the symmetry group is known as a non-abelian group.

In the local symmetry of the theory we consider infinitesimal local phase transformations. In this transformation ordinary derivative acting on the free Dirac field does not transform covariantly. So we define covariant derivative.

To define covariant derivative we require a new field called Gauge field which is coupled with field. The number of gauge fields equal to dimension of group G. Because gauge field depend upon generator of the group(Lie group).

Now covariant derivative transform covariantly under infinitesimal local transformations.

Rahul Aggrawal

Researcher@Ramanujan Shodh Sansthan


Rahul Aggrawal

I am a teacher and a theoretical physicist. Physics gives me pleasure and teaching physics gives me stable happiness. For More info visit www.rahulaggrawalphysics.blogspot.com

2 Comments

Hiroji kurihara · April 30, 2019 at 1:46 am

Lorentz contraction

Plain waves of light (wavelength is constant) are coming from the upper right 45 degrees. Two bars of the same length are moving to the right and the left at the same speed. The number of waves hitting the bars is the same. Lorenrz contraction is unthinkable.

Arpit nailwal · January 19, 2020 at 7:45 am

It was a nice article

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