Why are applications of group theory needed in the study of Physics in general?

The main motivation which results in exploiting the abstract mathematical concept of "Group theory" in physics stems from the connection between " Conservation Laws"

and "Symmetry" realized in the well known "Noether's theorem". The correspondence between invariance under a transformation of coordinates(internal or external) and conservative physical quantities (or in a quantum sense; observables) leads to this idea that these transformations could be expressed in the language of group concept (continuous or discrete). So the strength of group theory previously used in solving of the problems in mathematical field i.e. differential and polynomial equations ( Galois and Lie researches) shifted to a new era in physics, from relativity to quantum ( field) theories and crystallography.

interestingly the group properties are definitely matches the dynamical concepts in physical world, roughly:

No change~ identity ; Reversibility ~ existence of inverse ; Additive changes~ associativity etc.

Therefore these properties are inherent of most Dynamical processes in physics and this makes this theory as a mainstay of modern physics. The symmetries are the keys of physical world and such groups which comprise the more symmetries ( such as U(1)×SU(2)×SU(3)) could interpret more phenomena or existing particles.

Hello: This discussions are quite interesting!

Rolf has mentioned transformation properties and geometric shapes.

Alexis has mentined symmetries.

Manouchehr mentions Noether's theorem, connecting symmetries

of the action under changes in generalized coordinates, to

conservation laws.

As I understand transformation properties are related to

space-time coordinate change, such as translation, rotation,

time evolution, they encode changes in a system during

changes in space time co-ordinates.

Noether's theorem identifies properties not changing during

transformations, or in other words conserved quantities, such

as momentum,angular momentum, total energy. Geometric

shapes of objects such as molecules can be described by

group properties. Here also transformation is quite handy.

For example a equilateral triangle is unchanged by a rotation

of 2 pi/3 about an axis passing through centroid.

Let us continue the discussion, cheers !

Dear Biswajoy,

Group theory enters inevitably wherever we have symmetries. The more fundamental question would therefore be:

"WHY WE HAVE SYMMETRIES IN NATURE?"

Enjoy answering this one all ye here on this thread.

Cheers,

Rajat

Dear Rajat

the new question ,in my opinion is equal to :

"The physics of any situation should be independent of the related coordinate choice"

this proposition reflects in any mind ro as an " Axiom" and respect to this axiom translation,rotation etc of coordinates will not impose any change in the physics of the problem , Simply this could be interpreted as "Symmetry" in physical sense.Therefore accepting the action (or Lagrangian) as the the functional determining the physics of any phenomena, existence of symmetry in nature will be equivalent to the existence of conservative quantities due to Noether's theorem. .

Let us go back to the original discussion on group theory.

From the discussions above, it seems to me that the first

step is to get all transformations which keeps a system

invariant. These transformations forms a set. The set may

be finite or infinite depending on number of transformations

which has the property of keeping a system unchanged.

Next step is to verify whether this set forms a group or not.

If it forms a group then we call it a "symmetry group". If it

doesn't then all group properties are not satisfied by this

set of operations.

Very good point Biswajoy.

Can you give an example of a set of symmetry operations on a physical system not orming any "symmetry group?"

Regards,

Rajat

Dera Amiri,

Coordinates in your reply do not necessarily mean space-time coordinates, right?

Regards,

Rajat

Rajat

The point is closure property of the group. Sometimes we may overlook

one or two symmetry operations. Then closure property will not be

satisfied. This is a cross check for counting number of symmetry

operations. I have done it for symmetries of an equilateral triangle

a few years ago.

The group was D_3. I will write an elaborate post in this thread

because you are interested. Thanks,

By the way, sticking to the main question, Murray Gell-Man has

applied group theory to Physics in another way. For the classification

of elementary particles.

Dear Rajat

Sure!

As i mentioned in my previous note, invariance of Lagrangian especially in modern field theories take into account the "internal coordinates such as : Isospin, hypercharge, Barionic number etc exploited to classify and predict the particles in multlplets, like Leptons, Hadrons, as well as gauge invariant theories that initiates from Lorentz gauge in classic electrodynamic and continued as Yang- Mills theories equipped with more complex unitary groups.

Response to latest note of Biswajoy I remind that the closure property is just the same things as we see in Linear algebra and vector spaces where the addition of any two elements is another element. this facilitates the Representation of a group in a suitable setting for our physical problem. All representations ( Reducible or Irreducible) up to my knowledge based on the Representation in Linear algebras (Matrices, spinors, and recently Twistors). and this is the main reason for application of " closure property"

Regards

I am glad that Manouchehr brought in the discussion of representations.

Specially, matrix representations are often used. Rotation in two dimension

has a 2x2 matrix representation. In three dimensions of course it is

a 3x3 matrix representation. This area definitely has applications of

group theory

Further more I would also like to mention about group algebras

corresponding to Lie groups. This is another area of application.

What is the usefulness of studying group algebras ?

Dear Biswajoy

Let us extend our discussion to a bit more deep level.

I ask for a new question: What is the definite importance of "Moment Map" in the

action of a Lie group on a symplectic manifold and its application in physics?

Alright Manoucher:

Please give us meanings of "Moment Map" and "Symplectic

Manifold". For the purpose of this discussion which is at the

basic level, we will appreciate simple descriptions such as,

"A Lie group is a manifold obeying the group properties. The

manifold must be smooth. The additional condition which a

Lie group must satisfy is that the group operations are

differentiable."

Also, we will appreciate, a comment on what

are applications of group algebras in Physics.

Dear Biswajoy,

You have evaded my question. Are there any physical problems with symmetries that do not form a group? Please reply with a concrete example.

Regards,

Rajat

Oscar Castillo-Felisola

Dear all:

One goal of physics is to find "Laws of Nature", which should be valid to "equivalent" observers (inertial observers).

Thus, if you describe a phenomenum with a certain (differential) equation, the equation must be invariant under the set of transformations which relate our equivalent observes.

Note that the transformations MUST satisty:

- associativity,

- inverse of each transformation

- a non-trasformation at all

- and closure... if I transform after a transformation, the result is also a transformation between equivalent observer,

i.e., the transformations form a group.

Therefore, from the begining groups theory is important for physics. The rest are more advance situations were group theory makes our life easier: Noether, Lagrangian, Spectrum, etc... but the truth is GROUP theory entered to the game long before we noticed!

Artur Sergyeyev

Let me provide a point of view which is perhaps more geometrical than physical but actually is closely related to the latter (e.g. the symmetry groups of differential equations or systems mentioned by @Oscar can, roughly speaking, be treated as transformation groups acting on the set of (smooth) solutions of differential equations or systems). Namely, groups originally arose as groups of transformations acting on certain geometric object (see e.g. http://en.wikipedia.org/wiki/Group_action). Transfomations leaving a given geometric object invariant, naturally satisfy the closure axiom (a transformation being composition of two transformations leaving the object invariant again has the same property) and the associativity condition (as it is satisfied by composition of mappings). The existence of the inverse transformation is also fairly typical. Finally, one always has the identity transformation which obviously leaves the object invariant. I hope this helps :)

Rajat.

My point is that as a first step one has to collect all symmetry

elements and then one has to check that whether this set

forms a group or not. We know that all groups are sets but

all sets are not groups. I do not know of any theorem which

states that symmetry elements must form a group. I guess in

most of the cases they do form a group. I cannot think of

any counter example right now. If anybody in this forum knows

of a counter example, please let us know. Thanks in advance.

Arthur.

Existence of inverse is guaranteed if identity (do nothing) and

closure are guaranteed. Do you agree with me ? Please

let me know.

Oscar.

Your point is that set of transformations relating two equivalent

observers must satisfy group axioms. This must me a

continuous group. Please let us know the formal name of

this group.

Oscar Castillo-Felisola

@Biswajoy:

Existance of inverse is not guarantee! Take the integer numberes with the product,

(Z, * ), it is closed and have identity... but no inverse for numbers other than 1 and -1.

If you're considering equivalent observers, the group is Galilean group for non-relativistic theories or Poincare for relativistic ones. However, group theory goes far beyond this!

Cheers

Before talking about ``symmetry'', it might be useful to talk of ``transformation''. Then the idea would be that, by applying a transformation to a state of a (physical) system, for instance, one would expect to obtain another state of the ``same'' system. In that case, I'd guess that one is led quite quickly to the conclusion that the set of such transformations does have the properties that define a group. So, if one now imagines the situation where such a transformation is a ``symmetry''-e.g. some quantity remains invariant-then one is, similarly, led to the conclusion that the transformations that have, in addition, *such* a property, will define a group.

Oscar:

Yes, existence of identity and closure does not

guarantee existence of inverse. It is an independent

axiom, which does not depend on other axioms.

I could not understand the product (z,*) of the set of

integers. Can you please explain this product.

Robert:

We know the concept of elements as building

blocks. For example natural elements constitute

our surroundings, nuclear particles are made of

elementary particles and so on. By symmetry

element I meant a member of the set of symmetry

transformations. Is it possible to give it a precise

definition ?

Dear All,

I agree with Stam that the set of symmetry operations of a physical system will surely form a group. Counter-examples are welcome.

Regards,

Rajat

The ``counter-examples'' appear when limits must be taken. Then it is not obvious that the limiting procedure preserves the group property-and ``anomalies'' may appear-that's what this term *means*. In that case the physical system, defined through the limiting procedure, does *not* enjoy this symmetry. So the transformations always form a group-it's the physical system that has properties that aren't, in this way, compatible with the group property of the set of transformations. The reason this way of expressing things is more useful, than giving up the group property, is that it is, nonetheless, possible to establish relations between physical quantities, that express just how the particular symmetry is (or isn't) satisfied-and these can be experimentally tested (or in computer simulations).

Anomalies arise when symmetries of a classical theory is

absent after quantization. Traditionally anomalies were studied

in the context of quantum field theories, however, there are

examples in simple quantum mechanics problems also,

Ref:

Coon, Holstein, Am. J. Phys. 70 (2002) 513–519;

Braaten, Hammer, Annals Phys. 322 (2007) 120–163.

In supersymmetric theories anomaly is sometimes used

to break supersymmetry.

Ref:

http://www.sciencedirect.com/science/article/pii/S0550321399003594

jean claude Dutailly

Symmetries are a decptive concept.

Actually groups appear in Physics (or any other scientific theory) because of the Principle of relativity : the laws should not depend on the observer. So when two obeservers measure a phenomenon using variables which are related (such as a vector measured in two different frames) there should be a relation between the measures. When this relation depends on a parameter, which an element of a group, we have a representation of the group. These structures are well known and so, from the structural features of the model which is used one can identify important properties. This is the starting point of gauge theories in theoretical physics.

@ jean claude Dutality

I agree that groups appear because of relativity principle.

In particle Physics a number of gauge theories have been

developed. Gauge symmetries are "internal symmetries"

they are not "space-time" symmetries. Gauge theories

can be divided into local gauge theories and global gauge

theories. Weinberg Salam model is a local gauge theory.

QCD is also a local gauge theory. The first is invariant

under SU(2)_L x U(1)_Y transformation while the second

is invariant under SU(3) color transformations.

Because you have talked on representations in your

last post; please elaborate on representations further.

Thank you,

Ref:

Gauge Theories Of The Forces Between Elementary Particles

Gerard 't Hooft (Utrecht U.). Jun 1980. 27 pp.

Published in Sci.Am. 242N6 (1980) 90-116

Here is an article on this subject by 't Hooft

http://www.staff.science.uu.nl/~hooft101/gthpub/GaugeFields_75.pdf

jean claude Dutailly

To Bisjawoy

Of course the principle of relativity appears :

in a change of local frame in the fiber bundle picture (of which GR is a special case)

in a change of global gauge in the simplified framework of SR

in a change of geometric gauge (space time), lonked to gravitation

in a change of gauge for the measure of other fields (what is called internal symmetry).

In a the last two cases actually the gauge is linked to a strucure of principla bundle, and the field is a connection.

What needs further understanding, at a conceptual level, is the meaning of a gauge which is, by definition, the expression of the field. A field varies from a point to another, so what is a "fixed gauge" ?

For the implementation of gauge theories you can have a look at my paper about Particles and fields on this site.

@jean.

Let us discuss the kinetic term of a non-abelian

gauge theory.

While writing kinetic terms we have to sum over

space time indices and also gauge index independently.

We write F^a mu nu F^a_mu nu. Here we can display

space-time indices and gauge indices distinctively.

Do you agree with me ? Here summing over space time

indices make it a Lorentz scalar whereas summing over

gauge indices make it a gauge singlet.

Dear Claude,

Do you mean to say that symmetries are not objective properties and that they are their because there can be more than one observers (choice of frames) for the description.

My opinion is that, while your point of view (call it active view) is correct, there can be the other point of view (passive) where the frame remains fixed and the system undergoes the symmetry transformation.

So it is not true that Relativity is essential to Symmetries, but it is certainly one of the two ways of looking at symmetries.

Regards,

Rajat

jean claude Dutailly

what I say is just that the wprd symmetry is not accurate, even if it is commonin Physics. What we have is a transformation law, and a mathematical object is symmetric if it is invariant under some transformation. And the distinction active / passive has no special interest. A bit of tradition does not hurt, but it is time to grow up !

Of course what can, also, occur, is that, upon realizing this limiting procedure, only a subgroup of the original group of symmetry transformations remains compatible with the limiting procedure. Then only the transformations, that belong to this subgroup are realized as symmetries.

In the example about the kinetic term the statements require qualification, because how the indices are contracted is not specified. If the spacetime indices are contracted with the Minkowski metric, then the resulting quantity is, indeed, a Lorentz scalar; the ``contraction'' over the group indices is realized by the the trace over the group generators. That is what renders it *invariant* under gauge transformations.

This is standard material on group theory and Yang-Mills fields. To go into the technical details, cf. http://www.math.ias.edu/qft

I guess I have to specify what is the index `a' and its possible

values.

For a SU(N) gauge symmetry the index `a' runs from

1 to N^2-1. This is because gauge bosons are in

adjoint representation. For SU(3) color group

the index a will have values 1,2...8.

In other words after contracting space-time indices

using Minkowski metric, one can form an 8x8 matrix whose

elements are

F^a_ mu nu F^b mu nu

a=1,2,...8 b=1,2,...8

then take trace of this matrix.

Christopher Eltschka

When we formulate the laws of physics, we make use of "objects" which are not really physical objects, but constructions we need to describe the physics. For example, to describe objects in space and time, we introduce coordinates, or to describe systems of several identical particles, we sometimes assign numbers to the particles (so we can write down e.g. the wave function in the coordinates of particle 1 and particle 2). Since the actual physics does not depend on those artificial constructs (after all, you'll not find a coordinate system in physical space and time, or labels at individual electrons), our equations must not depend on any aspect of the construct that is not derived from/related to physical properties, or else they cannot correctly describe reality. That is, the equations must be invariant under change of those artificial properties. Such changes always form a group, and group theory effectively tells us what our equations have to look like if they shall have that symmetry.

Note that this is not strictly about dependence on an *observer*; in relativity, we like to define our coordinates in relation to an observer, since then we can relate the numbers directly to what the observer observes. But that's not strictly necessary. We can define coordinates which correspond to no observer, and the equations still have to work correctly.

I would like to put down a note on matrix representations

of symmetry operations.

Closure, associativity, identity, existence of inverse are concepts

in relation to the definition of a binary combination rule or a

"product". For symmetry operations the product is defined as

follows. Make first operation o1, and then make second

operation o2. The system goes to a new state. Now consider

an operation o3 which takes initial state to the new state.

In the language of symmetry operations o3=o1*o2

But now suppose we can map operations to a group

of matrices. Then symmetry operation o1 corresponds

to a matrix m1. Symmetry operations o2 and o3 to

matrices m2 and m3. Then m3=m1*m2

The advantage here is that we no longer have to take

a system and do symmetry operations on it again and

again. We can simply multiply matrices using matrix

multiplication rules which is known to every-body.

I would like to know about other advantages of using

representations.

That groups admit matrix representations is a statement about mathematics, not physics. The mathematical question has been fully explored, with regard to ``classical'' groups.(``Quantum'' groups are a different matter.)

I don't think this is what's under discussion here. For physics, any such representations imply constraints on the physical system, at the classical level and at the quantum level. A lot is known about these issues, too and is the subject of group theory applied to physics, material that can be found in many books and courses. The representation however is just a tool-it can't determine the physics, just describe it and, *given* some specific physical situation, lead to some predictions-but it can't conjure up a physical situation.

The most part of physical systems represent the "Linear nature" i.e. sysyems could be modeled via a "Vector space", these vector spaces include the primary tools in all part of physics, examples in classic and quantum are well known "Hilbert space" in quantum models and "4- vector space" in special relativity , "Tensor space"

in General relativity, "spinor and twistor space" in particle physics etc. Consequently the "Linear" nature of physical systems is the reason for working matrix representation which as a tool in mathematics enabling us to model the complex phenomena. The "Linearity" in Nature leads to the efficiency and applicability of concepts such as vector,matrix,tensor and spinor. In most models this Linearity turns out to be appeared on Tangent space of the problem manifold or on fiber bundle of a Base space. Language to define "Physics" restricted to these spaces which originated from abstract concepts in mathematics, in this sense there is not a distinct border between "physics" and these abstract concepts. How can we describe a photon other than a vector field A_μ ?

@Manoucher Amiri

I also think that at a deeper level there is no distinction between Physics

and Mathematics.

But see that Einstein field equations are non-linear. I am quoting,

"The nonlinearity of the EFE distinguishes general relativity from

many other fundamental physical theories. For example, Maxwell's

equations of electromagnetism are linear in the electric and magnetic

fields, and charge and current distributions (i.e. the sum of two solutions

is also a solution); another example is Schrödinger's equation of

quantum mechanics which is linear in the wavefunction."

source:

http://en.wikipedia.org/wiki/Einstein_field_equations

Then the question naturally arises whether the corresponding

action is invariant under symmetry transformations which are

linear in field variables defining the action.

On the other hand, Einstein field equations are linear

(first power) if you express in terms of tensors.

G_{mu nu}= (8 pi G/c^4) T_{mu nu}

Then do you mean that these tensors will have a set of

linear transformations which will keep action invariant.

@ Biswajoy Brahmachari

I agree EFE are a well known exception in the long list of Linear equations in theoretical physics. Existence of non-linear differential equations and singularities occurring in the physical systems is a pitfall that imposes some other methods to evaluate such systems ,Chaotic behavior, catastrophe theory ...etc are emerged from this viewpoint. However the GR related action (i.e. Einstein Hilbert action) should be Lorentz invariant, in this sense Tensor transformations are linear, I mean that all Tensors (not pseudo-tensors) transform by terms that can be exactly expressed in Matrix froms (recall that the transformation terms are in the form: (∂(x_i ) ̅)/(∂x_j ) which connects the old and new coordinates (similar to Jacobi matrix). therefore we can label the " covariant" for these type of equations.

Oscar Chavoya-Aceves

@Jean Claude

I understand and share your point. I will appreciate though if you go deeper on your statement: "When two observers measure a phenomenon using variables which are related (such as a vector measured in two different frames) THERE SHOULD BE a relation between the measures."