An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) Updated 2024-12-15 +Created 1970-01-01
This does not seem to go deep into the Standard Model as Physics from Symmetry by Jakob Schwichtenberg (2015), appears to focus more on more basic applications.
But because it is more basic, it does explain some things quite well.
A suggested at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles", it appears that in the Standard Model, the behaviour of each particle can be uniquely defined by the following five numbers:
E.g. for the electron we have:
- mass:
- spin: 1/2
- electric charge:
- weak charge: -1/2
- color charge: 0
Once you specify these properties, you could in theory just pluck them into the Standard Model Lagrangian and you could simulate what happens.
Setting new random values for those properties would also allow us to create new particles. It appears unknown why we only see the particles that we do, and why they have the values of properties they have.
The Dirac equation can be derived basically "directly" from the Representation theory of the Lorentz group for the spin half representation, this is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) 6.3 "Dirac Equation".
The Diract equation is the spacetime symmetry part of the quantum electrodynamics Lagrangian, i.e. is describes how spin half particles behave without interactions. The full quantum electrodynamics Lagrangian can then be reached by adding the internal symmetry.
As mentioned at spin comes naturally when adding relativity to quantum mechanics, this same method allows us to analogously derive the equations for other spin numbers.
Bibliography:
Deriving The Dirac equation by Andrew Dotson (2019)
Source. The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":
- spacetime symmetry: act with the Poincaré group on the Four-vector spacetime inputs of the field itself, i.e. transforming into
- internal symmetry: act on the output of the field, i.e.:
From defining properties of elementary particles:
- spacetime:
- internal
From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.
Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".
Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.
Another important way to think about Lie algebras, is as infinitesimal generators.
Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.
To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.
- the dimension
- the Lie bracket
As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.
Bibliography:
Subgroup of the Poincaré group without translations. Therefore, in those, the spacetime origin is always fixed.
Or in other words, it is as if two observers had their space and time origins at the exact same place. However, their space axes may be rotated, and one may be at a relative speed to the other to create a Lorentz boost. Note however that if they are at relative speeds to one another, then their axes will immediately stop being at the same location in the next moment of time, so things are only valid infinitesimally in that case.
This group is made up of matrix multiplication alone, no need to add the offset vector: space rotations and Lorentz boost only spin around and bend things around the origin.
One definition: set of all 4x4 matrices that keep the Minkowski inner product, mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) page 63. This then implies:
Ciro Santilli's favorites so far:
Bibliography of the biliograpy:
- physics.stackexchange.com/questions/8441/what-is-a-complete-book-for-introductory-quantum-field-theory "What is a complete book for introductory quantum field theory?"
- www.quora.com/What-is-the-best-book-to-learn-quantum-field-theory-on-your-own on Quora
- www.amazon.co.uk/Lectures-Quantum-Field-Theory-Ashok-ebook/dp/B07CL8Y3KY
Recommendations by friend P. C.:
- The Global Approach to Quantum Field Theory
- Lecture Notes | Geometry and Quantum Field Theory | Mathematics ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/lecture-notes/
- Towards the mathematics of quantum field theory (Frederic Paugam)
- Path Integrals in Quantum Mechanics (J. Zinn–Justin)
- (B.Hall) Quantum Theory for Mathematicians (B.Hall)
- Quantum Field Theory and the Standard Model (Schwartz)
- The Algebra of Grand Unified Theories (John C. Baez)
- quantum Field Theory for The Gifted Amateur by Tom Lancaster (2015)
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , that has the same properties as the group.
Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).
Each such matrix then represents one specific element of the group.
This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.
Or more precisely, mapping each group element to a linear map over some vector field (which can be represented by a matrix infinite dimension), in a way that respects the group operations:
As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)
- page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
- 3.6 classifies the representations of . There is only one possibility per dimension!
- 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
- www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
- www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.
Physics from Symmetry by Jakob Schwichtenberg (2015) page 66 shows one in terms of 4x4 complex matrices.
More importantly though, are the representations of the Lie algebra of the Lorentz group, which are generally also just also called "Representation of the Lorentz group" since you can reach the representation from the algebra via the exponential map.
Bibliography:
- Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.7 "The Lorentz Group O (1, 3)"
I like relativistic quantum mechanics.
Best mathematical explanation: Section "Spin comes naturally when adding relativity to quantum mechanics".
Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:
- spin 0: representation
- spin half: representation
- spin 1: representationAs usual, we don't know why there aren't elementary particles with other spins, as we could construct them.
TODO understand a bit more intuitively.
- Physics from Symmetry by Jakob Schwichtenberg (2015) page 72
- physics.stackexchange.com/questions/172385/what-is-a-spinor
- physics.stackexchange.com/questions/41211/what-is-the-difference-between-a-spinor-and-a-vector-or-a-tensor
- physics.stackexchange.com/questions/74682/introduction-to-spinors-in-physics-and-their-relation-to-representations
- www.weylmann.com/spinor.pdf
Why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics? Updated 2024-12-15 +Created 1970-01-01
Physicists love to talk about that stuff, but no one ever has the guts to explain it into enough detail to show its beauty!!!
Perhaps the wisest thing is to just focus entirely on the part to start with, which is the quantum electrodynamics one, which is the simplest and most useful and historically first one to come around.
Perhaps the best explanation is that if you assume those internal symmetries, then you can systematically make "obvious" educated guesses at the interacting part of the Standard Model Lagrangian, which is the fundamental part of the Standard Model. See e.g.:
- derivation of the quantum electrodynamics Lagrangian
- Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 7 "Interaction Theory" derives all three of quantum electrodynamics, weak interaction and quantum chromodynamics Lagrangian from each of the symmetries!
One bit underlying reason is: Noether's theorem.
Notably, axelmaas.blogspot.com/2010/08/global-and-local-symmetries.html gives a good overview:so it seems that that's why they are so key: local symmetries map to the forces themselves!!!
A local symmetry transformation is much more complicated to visualize. Take a rectangular grid of the billiard balls from the last post, say ten times ten. Each ball is spherical symmetric, and thus invariant under a rotation. The system now has a global and a local symmetry. A global symmetry transformation would rotate each ball by the same amount in the same direction, leaving the system unchanged. A local symmetry transformation would rotate each ball about a different amount and around a different axis, still leaving the system to the eye unchanged. The system has also an additional global symmetry. Moving the whole grid to the left or to the right leaves the grid unchanged. However, no such local symmetry exists: Moving only one ball will destroy the grid's structure.Such global and local symmetries play an important role in physics. The global symmetries are found to be associated with properties of particles, e. g., whether they are matter or antimatter, whether they carry electric charge, and so on. Local symmetries are found to be associated with forces. In fact, all the fundamental forces of nature are associated with very special local symmetries. For example, the weak force is actually associated in a very intricate way with local rotations of a four-dimensional sphere. The reason is that, invisible to the eye, everything charged under the weak force can be characterized by a arrow pointing from the center to the surface of such a four-dimensional sphere. This arrow can be rotated in a certain way and at every individual point, without changing anything which can be measured. It is thus a local symmetry. This will become more clearer over time, as at the moment of first encounter this appears to be very strange indeed.
axelmaas.blogspot.com/2010/09/symmetries-of-standard-model.html then goes over all symmetries of the Standard Model uber quickly, including the global ones.