Is the Expression Valid in Demonstrating a Poincare Algebra Commutator?

In summary, the conversation discusses the equation ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0## and its relation to the Poincare algebra commutator. It is suggested that the equation can be shown to be true using the fact that partial derivatives commute and by renaming the indices. However, there is confusion about this approach due to the indices being free rather than dummy. It is suggested to consider a particular case where the indices are all different and one of the variables is set to 0.
  • #1
binbagsss
1,265
11

Homework Statement



Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0##

Homework Equations



I need this to be true to show a poincare algebra commutator.

We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute.

Where ##P_u=\partial_u##

The Attempt at a Solution


[/B]
I'm guess it's using that partial derivatives commute and renaming the ##v## and ##p## for each other, however I'm a bit confused with this, because, I know this isn't really the context here, but these are free indicies not dummy indices, i..e not an index summing over, so I'm a bit unclear how you could simply rename.

Many thanks
 
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  • #2
binbagsss said:

Homework Statement



Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0##

Homework Equations



I need this to be true to show a poincare algebra commutator.

We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute.

Where ##P_u=\partial_u##

The Attempt at a Solution


[/B]
I'm guess it's using that partial derivatives commute and renaming the ##v## and ##p## for each other, however I'm a bit confused with this, because, I know this isn't really the context here, but these are free indicies not dummy indices, i..e not an index summing over, so I'm a bit unclear how you could simply rename.

Many thanks

Free indices means that it must be true for every possible choice of [itex]u,p,v[/itex], and for all possible values of [itex]x_p[/itex] and [itex]x_v[/itex]. So try the particular case where [itex]u,p,v[/itex] are all different indices, and [itex]x_p = 0[/itex], and [itex]x_v \neq 0[/itex].
 

Related to Is the Expression Valid in Demonstrating a Poincare Algebra Commutator?

1. What is the Poincare Algebra?

The Poincare Algebra is a mathematical structure that describes the symmetries of space and time in the theory of special relativity. It is a Lie algebra, which means it is a vector space equipped with a bilinear operation called the Lie bracket. In physics, the Poincare Algebra is used to study the symmetries of physical systems and to understand the behavior of particles and fields.

2. What are the elements of the Poincare Algebra?

The elements of the Poincare Algebra are generators of transformations that preserve the form of the equations of motion in special relativity. These include the generators of translations in space and time, rotations in space, and boosts in velocity. In total, there are 10 generators of the Poincare Algebra, which can be written as 6 rotations and 4 boosts.

3. How is the Poincare Algebra related to the Lorentz Group?

The Poincare Algebra is closely related to the Lorentz Group, which is the group of transformations that preserve the Minkowski metric in special relativity. The generators of the Poincare Algebra can be used to construct the elements of the Lorentz Group, and the Lie bracket operation of the Poincare Algebra corresponds to the commutator operation of the Lie group.

4. What is the significance of the Poincare Algebra in physics?

The Poincare Algebra is a fundamental mathematical structure in physics, particularly in the fields of special relativity and quantum field theory. It allows us to study the symmetries of physical systems and understand how particles and fields behave under these symmetries. It also plays a crucial role in the formulation of gauge theories, which are essential in modern physics.

5. Are there any applications of the Poincare Algebra outside of physics?

Yes, the Poincare Algebra has applications in mathematics, particularly in the fields of group theory and Lie algebras. It has also been used in computer graphics and image processing to study the symmetries of shapes and patterns. Additionally, it has been used in linguistics to study the symmetries of language structures.

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