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JD_PM

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- TL;DR Summary
- I'd like to gain more insight on the Lorentz Group, its most important subgroups and must-know examples in Physics. At the moment I only know a handful: the orthogonal group ##O(3)##, the orthogonal group ##SO(3)## and the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]

This thread is motivated by samalkhaiat's comment here

I know that the Lorentz Group is formed by all matrices that satisfy

$$\eta = \Lambda^{T} \eta \Lambda \tag{1.1}$$

Which is equivalent to

$$\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma} \tag{1.2}$$

If we add no more restrictions and define the group under the inner product we end up with the group ##O(1,3)##.

I also know that all orthogonal matrices must satisfy ##1_3 =R^T 1_3 R##. They include not only rotational matrices but also space and time reversals, also known as parity transformations (i.e. [itex](x^{0} , x^{i}) \to (x^{0} , -x^{i})[/itex]) and [itex](x^{0} , x^{i}) \to (-x^{0} , x^{i})[/itex] respectively)

I have some questions:

1) Is the Lorentz group always defined under the inner product? The only examples I know are; for instance: the orthogonal group ##O(3)##, the orthogonal group ##SO(3)## and the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]

2) I've read that ##O(1,3)## has one positive and one negative eigenvalue of its defining symmetric matrix. I do not see why, how could we prove it?

3)

If I am not mistaken, this is because [itex]SO^{\uparrow}(1,3)[/itex] requires matrices to fulfil not only ##1_3 =R^T 1_3 R## condition but also that its determinant must be 1. The later condition is not fulfilled by matrices representing parity transformations (spatial, time and space-time reversals), but how can I prove this?Sources:

SpaceTime & Geometry by Carroll, pages 12,13,14

vanhees71 QFT manuscript, section 3.1.

Any help is appreciated.

Thank you

samalkhaiat said:That isneithercontinuousnorconnected Lorentz transformation. It is adiscretespace-time reversal [itex](x^{0} , x^{i}) \to (-x^{0} , -x^{i})[/itex]. Space reflection [itex](x^{0} , x^{i}) \to (x^{0} , -x^{i})[/itex]; time reversal [itex](x^{0} , x^{i}) \to (-x^{0} , x^{i})[/itex] and space-time reversal formdisjoint subsetsand are notcontinuously connectedto the identity. In English, [itex]x \to – x[/itex] does not belong to the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]. By the “Lorentz group”, we always mean the real semi-simple Lie group [itex]SO^{\uparrow}(1,3)[/itex].

I know that the Lorentz Group is formed by all matrices that satisfy

$$\eta = \Lambda^{T} \eta \Lambda \tag{1.1}$$

Which is equivalent to

$$\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma} \tag{1.2}$$

If we add no more restrictions and define the group under the inner product we end up with the group ##O(1,3)##.

I also know that all orthogonal matrices must satisfy ##1_3 =R^T 1_3 R##. They include not only rotational matrices but also space and time reversals, also known as parity transformations (i.e. [itex](x^{0} , x^{i}) \to (x^{0} , -x^{i})[/itex]) and [itex](x^{0} , x^{i}) \to (-x^{0} , x^{i})[/itex] respectively)

I have some questions:

1) Is the Lorentz group always defined under the inner product? The only examples I know are; for instance: the orthogonal group ##O(3)##, the orthogonal group ##SO(3)## and the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]

2) I've read that ##O(1,3)## has one positive and one negative eigenvalue of its defining symmetric matrix. I do not see why, how could we prove it?

3)

samalkhaiat said:[itex]x \to – x[/itex] does not belong to the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]. By the “Lorentz group”, we always mean the real semi-simple Lie group [itex]SO^{\uparrow}(1,3)[/itex].

If I am not mistaken, this is because [itex]SO^{\uparrow}(1,3)[/itex] requires matrices to fulfil not only ##1_3 =R^T 1_3 R## condition but also that its determinant must be 1. The later condition is not fulfilled by matrices representing parity transformations (spatial, time and space-time reversals), but how can I prove this?Sources:

SpaceTime & Geometry by Carroll, pages 12,13,14

vanhees71 QFT manuscript, section 3.1.

Any help is appreciated.

Thank you