Groups in Physics: What are they Used For?

[Schreiberdk]

Hi there PF

What are groups, and what are they used for in physics? For example if you look at QED, http://en.wikipedia.org/wiki/Quantum...cs#Mathematics , it is said here that QED is a abelian gauge theory with symmetry group U(1). What is this symmetry group, and what is it used for when you do calculations within the theory?

\Schreiber

 

[cosmik debris]

A group is an algebraic structure that incorporates axioms and operations that apply to its members. A symmetry group tells you what operations produce invariants and so gives you quantities which are conserved via Noether's theorem.

 

[Kevin_Axion]

The $$U(1)$$ gauge group is a rotational symmetry group of the complex unit circle. It's representation is from Euler's equation:

$$e^{i\theta} = cos\theta + isin\theta$$

where $$\theta$$ is some angle.

Also the special orthogonal group $$SO(2)$$ is isomorphic ("related") to the $$U(1)$$ symmetry group. $$SO(2)$$ is just a rotation in $$\Re^2$$ and is represented by the 2 x 2 matrix:

$$SO(2) = \begin{pmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \\ \end{pmatrix}$$

 

[Schreiberdk]

So one uses the SO(2) symmetry for vectors invariant under rotations, while U(1) is used for scalars invariant under rotation?

But what does this give a physical theory? and how does it get incorporated when you do calculations with the theory?

 

[espen180]

According to Noether's theorem, a symmetry in nature corresponds to a conserved quantity. For example, the homogeneity of space (invariance under spatial translations) corresponds to conservation of the linear momentum vector.

 

[Kevin_Axion]

So one uses the SO(2) symmetry for vectors invariant under rotations, while U(1) is used for scalars invariant under rotation?

But what does this give a physical theory? and how does it get incorporated when you do calculations with the theory?

No, a scalar rotation doesn't make any physical sense.

It gives the theory symmetry, in Physics there are always conserved quantities and $$U(1)$$ conserves electric charge.

 

[Schreiberdk]

May i then ask how this is so ? It does not seem obvious to me :) If there is a simpler example than QED, I would probably understand that better, since my physics knowledge only goes around classical physics + abit of quantum mechanics :)

 

[espen180]

http://en.wikipedia.org/wiki/Symmetry_(physics)#Mathematics_of_physical_symmetry

 

[Kevin_Axion]

May i then ask how this is so ? It does not seem obvious to me :) If there is a simpler example than QED, I would probably understand that better, since my physics knowledge only goes around classical physics + abit of quantum mechanics :)

Okay so imagine we have a unit vector $$\beta$$:

$$\beta = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix}$$
So, $$\beta$$ has magnitude 1 (unit vector) and lies along the x-axis.
Now imagine we have a group $$SO(3)$$:

$$SO(3) = \begin{pmatrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$

$$SO(3)$$ is a 3 x 3 matrix that is the symmetry group of a sphere. Now $$SO(3)$$ is known as a transformation matrix as when it is applied to a vector like $$\beta$$ it rotates it by an angle $$\theta$$ counter-clockwise.
So imagine we take the the group $$SO(3)$$ and apply it's matrix representation to the vector $$\beta$$ and it rotates it by some angle $$\theta$$ counter-clockwise what property of the vector is conserved?

Well, it's length (magnitude) is conserved. If we use the magnitude formula:

$$||\beta|| = \sqrt{\beta \cdot \beta}$$

before the transformation we know since it is a unit vector it's magnitude is 1. After the transformation if we use the magnitude formula again we still get a magnitude of 1 i.e length is conserved under any $$SO(3)$$ transformation.

 

[element4]

$$SO(3) = \begin{pmatrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$

Just a small correction, you have only parametrized the subgroup of $$SO(3)$$ consisting of rotations in the x-y plane. Its the isotropy subgroup of the point $$\begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix}$$ on the sphere (this point is invariant under all these transformations). More generally, $$SO(3)$$ is the set of all real and orthogonal $$3\times 3$$ matrices with determinant 1.

 

[Kevin_Axion]

Just a small correction, you have only parametrized the subgroup of $$SO(3)$$ consisting of rotations in the x-y plane. Its the isotropy subgroup of the point $$\begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix}$$ on the sphere (this point is invariant under all these transformations). More generally, $$SO(3)$$ is the set of all real and orthogonal $$3\times 3$$ matrices with determinant 1.

Yea, thanks for the correction. I don't really know all of the complexities of $$SO(3)$$ as I haven't taken a linear algebra or group theory course. I only did one rotation on one plane for simplicities sake.

 

[A. Neumaier]

Hi there PF

What are groups, and what are they used for in physics? For example if you look at QED, http://en.wikipedia.org/wiki/Quantum...cs#Mathematics , it is said here that QED is a abelian gauge theory with symmetry group U(1). What is this symmetry group, and what is it used for when you do calculations within the theory?

Groups are sets of (theoretical) motions that one can perform within some system without altering certain properties. For example, in ordinary space, you can translate and/or rotate things without changing the distances between their parts - this defines the group of Euclidean motions. You can also squeeze objects - this changes distances but preserves the local (differential geometric) structure - this gives the group of diffeomorphisms.

Whatever structure you are considering, the group of transformations preserving it is its symmetry group.

 

[Schreiberdk]

Well, why is it so important? why not just say: This quantity is conserved? why does one have to use a mathematical group for it?

And do you even use the group, when you do calculations within the theory?

 

[A. Neumaier]

Well, why is it so important?

The point is that the same abstract group appears over and over again in very different realizations of symmetries. Thus their common properties need to be studied only once and can then be used everywhere.

Like with every mathematical concept. Having good concepts allows one to recognize connections much better, and to reuse existing techniques in new situations.

For example, rotations around a fixed point form a geometric group, which is the same as the group SO(3) of orthogonal 3x3 matrices. Rotations (as many other symmetries) can be performed in arbitrarily tiny steps, giving rise to the concept of an infinitesimal symmetry. Associated to these are Lie algebras.

One doesn't want to do this construction again and again for each new case of symmetries - so one needs a general theory.

The generators of the Lie algebra of SO(3) happen to be essentially the same objects as the components of angular momentum. The Lie algebra is the same as the one you get when you study the symmetries of a single qubit.

The representations of the Lie algebra tell you something about the spectrum of certain quantum observables. The representation theory of the group SU(3) lead to the discovery of a new elementary particle - the Omega minus.

One doesn't what to do representation theory for each single case, and not several times for each concrete occurrence of the same group or Lie algebra. Thus one needs a common theory.

Many connections between seemingly unrelated things come through groups.
Indeed, groups (and Lie algebras) are so fundamental that one cannot do modern physics without it.

why not just say: This quantity is conserved?

''conserved'' has in physics a very special meaning, and is not appropriate here.

why does one have to use a mathematical group for it?

Because the theory of groups is so well developed that trying to doing without it is a neglect of valuable resources.

And do you even use the group, when you do calculations within the theory?

Most calculations become far simpler when the symmetries can be exploited through an informed use of group theory. The spectrum of the hydrogen atom (Balmer lines, etc.) comes out most nicely using group theoretic calculations - saving one lots of messy analysis.

 

[smith345]

According to Noether's theorem, a symmetry in nature corresponds to a conserved quantity. For example, the homogeneity of space (invariance under spatial translations) corresponds to conservation of the linear momentum vector.