Representations of Poincare group

  • #1

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I need to study in detail the rappresentations of the Poincare Group, i am interessed in the idea that particles can be wieved as irriducible representations of it.
Do you have some references about it?
 

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  • #2
George Jones
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A nice introduction is given in the book "Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics" by Sexl and Urbantke. See also chapter 21 from "Theory of Group Representations and Applications" by Barut and Raczka.
 
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  • #3
strangerep
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I need to study in detail the rappresentations of the Poincare Group, i am interessed in the idea that particles can be wieved as irriducible representations of it.
Ah, this is one of the deepest, most elegant concepts in all of physics. And group representation theory doesn't just apply to the physics of elementary particles -- it is widely pervasive.

As well as the references George mentioned, you can also try Weinberg vol 1, or perhaps Magiorre.

But it depends on your level of formal physics education so far. E.g., can you read and understand Ballentine's textbook on ordinary QM? Ch 7.1 in particular introduces one to the way the quantum restriction to half-integral angular momentum is found just by investigating the unitary irreducible representations of the ordinary spatial rotation group (with basically no other assumptions). Indeed, before diving into the Poincare group for the full relativistic case, one should understand the nonrelativistic quantum angular momentum case thoroughly (since it gets re-used in the full Poincare case).

One can even derive the basic non-relativistic H-atom energy spectrum from the symmetries of the equation for that problem (i.e., by requiring that the symmetry operators be represented unitarily), i.e., without having to solve the Schrodinger eqn.

Good luck!
 
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I am at the last year of specialization in theoretical physics, i have also followed in detail a course in group theory and i need a serious reference in order to study in detail the rappresentations of the Poincarè group
 
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Ah, this is one of the deepest, most elegant concepts in all of physics. And group representation theory doesn't just apply to the physics of elementary particles -- it is widely pervasive.
I just want to reinforce the above. It is very very deep and elegant - one of the greatest discoveries physicists and mathematicians have made.

There are two books I suggest, although they are not specifically about it, but rather stress the underlying importance of symmetry:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
https://www.amazon.com/dp/0801896940/?tag=pfamazon01-20

The first by Landau had a very deep effect on me.

Thanks
Bill
 
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George Jones
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I am at the last year of specialization in theoretical physics, i have also followed in detail a course in group theory and i need a serious reference in order to study in detail the rappresentations of the Poincarè group
I am not sure what you mean by "serious reference". Have you looked at the references already given in this thread and found that they are not adequate? If so, in what way are they not adequate? By "serious reference", do you mean "mathematically rigourous"?

Weinberg's treatment is nice, but maybe not rigorous. Duncan, in his book "The Conception Framework of Quantum Field Theory", gives a treatment similar to Weinberg's, but (I think) Duncan explains things better.

The second reference that I gave in post #2 is (close to being) mathematically rigourous, and it does a nice job in showing the role of relativistic wave equation in projecting onto invariant subspaces.

A mathematical overview of representations of the Poincare group is group is given in the book "Quantum Field theory: A Tourist Guide for Mathematicians" by Folland. For this, Folland gives "Lie Groups and Quantum Mechanics" by Simms and "The Geometry of Quantum Theory" by Varadarajan as references.
 
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A. Neumaier
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i need a serious reference in order to study in detail the representations of the Poincarè group
You find a lot of material (including references) in Chapter B1: The Poincare group of my theoretical physics FAQ.
 
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  • #8
With "serious references" i was only saying "rigorous from a matematical point of view" (i am Italian and my English could lead to misunderstanding).
Anyway thank you George, but unluckly i can' t find in internet the books that you linked to me, i have found only the Ballantine that linked Strangerep: yes, i can read it, but i can' t find what i am looking for.
Thanks A. Neumaier, i will take a look
 

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