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In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains fixed under some transformation. More importantly the laws of physics or equations governing those laws are symmetric under certain transformation of variables. The equations remain unchanged under these tranformations. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be explained using Lie groups while discrete symmetries are described by finite groups (see Symmetry group).
These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.
Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is known in mathematical terms as the Poincaré group, the symmetry group of special relativity. Another crucial example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important concept in general relativity.


Symmetries may be broadly classified as global or local. A global symmetry is one that holds at all points of spacetime, whereas a local symmetry is one that has a different symmetry transformation at different points of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates. Local symmetries play an important role in physics as they form the basis for gauge theories.

"we are not here because we are free , we are here because we are not free."


Gauge symmetry

Certain type of transformation when applied provides invariance for the lagrangian in field theory. Gauge transformations are transformations in which the Lagrangian is invariant under a continuous group of local transformations. An invariant is a model that holds true no matter the mathematical procedure applied to it. This continuous group shows some symmertry. This kind of symmetry is called gauge symmetry :

Gauge symmetry

Gauge symmetry

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Noether's theorem

All fine technical points aside, Noether's theorem can be stated informally
If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

Noether theorem

The symmetries are in the transfromations of the generalized coordinates q(k) and in turns , in Lagrangian L. The transformation can be a time reversal or other change , which will leave the lagrangian invariant. We are varying the langrangian and generalized coordinates but the quantity in the parenthesis does not vary over time. This is deduced from the principles of the extrema values of functions. The quantity in the parenthesis may be called "constant of motion". The constant of motion, in general" is the quantity which is conserved. Angular momentum is a constant of motion.



Noether theorem

"Continuous symmetry breaking"

Continuous symmetry breaking is a concept in theoretical physics. In the early stages of big bang there was this breaking up of symmetry. System or physical changes becomes asymmetric after this symmetry breaking.
In explicit symmetry breaking, if we consider two outcomes, the probability of a pair of outcomes can be different. By definition, spontaneous symmetry breaking demands the existence of a symmetric probability distribution—any pair of outcomes has the same probability. In other words, the underlying laws[clarification needed] are invariant under a symmetry transformation.
Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions can be explained by spontaneous symmetry breaking. Notable exceptions include topological phases of matter like the fractional quantum Hall effect.

Mexican hat potential

Consider a symmetric upward dome with a trough circling the bottom. If a ball is put at the very peak of the dome, the system is symmetric with respect to a rotation around the center axis. But the ball may spontaneously break this symmetry by sliding down the dome into the trough, a point of lowest energy. Afterward, the ball has reached some fixed point on the perimeter. The dome and the ball retain their individual symmetry, but the system does not.

symmetry breaking potenstial
In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian of a scalar field φ , which essentially dictates how a system should behave, can be split up into kinetic and potential terms,

symmetry breaking potenstial
for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of θ), this symmetry will appear to be lost, or "spontaneously broken".
In fact, any other choice of θ would have exactly the same energy, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.

Discrete symmetry

Discrete sysmmetry can be found in Dirac equation.
symmetry breaking potenstial

CPT Symmetry

CPT symmetry is the symmetry involving charge conjugation, parity and time reversal symmetries. C stands for the symmetry of charge conjugation. That is to say, if you change the charge of particle from positive to negative or from negative to positive the symmetry will be related to this charge conjugation. Similarly the symmetry of mirror image is called parity or mirror symmetry. If we stand beside a mirror our left hand is transformed into right hand and our right hand is transform into left hand. The transformed image is not the same and we can not say our mirror image is symmetrical. If the laws of physics is unchanged under this change of mirror reflection then the law can be said to preserve mirror symmetry. It is found that weak nuclear decay violates this mirror symmetry. It also violates the C or charge conjugation symmetry. And the time reversal symmetry is related to the reversal of time in opposite dir ection. Many laws are time symmetric . That means if you reverse the time the law remain the same. Gravity does not care if time runs forward or backward. It is time symmetric. The CPT symmetry can be explained with a diagram as follows:


symmetry breaking potenstial


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