Symmetries of Particles and Fields

Table of Contents

Metric tensor

  1. symmetries_of_particles_and_fields_cdbdfda567d42716e369cf1d2a5fb8bd61074f85.png has real eigenvalues symmetries_of_particles_and_fields_2f7de0bb72a7f95147202acfbb505010062a8b5a.png:

    symmetries_of_particles_and_fields_d84fec294678e0300928bddb7e8c01f689a8070d.png

  2. Rescale sucht that symmetries_of_particles_and_fields_a9b00451d845267083dd036c479ea5fe96249e45.png keeping symmetries_of_particles_and_fields_b18646cd706ddda6e056396c8b2a4e234455c661.png fixed. Rescaling follows

    symmetries_of_particles_and_fields_33d7b37ef9d6d8aeecce87fece644d19ede05453.png

    which we can do separately for each symmetries_of_particles_and_fields_2f7de0bb72a7f95147202acfbb505010062a8b5a.png, hence

    symmetries_of_particles_and_fields_2b6bb767721776109c64dea2444cbae685fd8848.png

    which we call the canonical basis.

Example: space time

symmetries_of_particles_and_fields_d2a9b859458df2a4d1f44bb51a2061277cdeae9d.png

such that

symmetries_of_particles_and_fields_d59cf7d3c315e0d16152d50650b4e03fdd58ef4d.png

Usually, we use the opposite signature metric, using symmetries_of_particles_and_fields_70e9cb9f0cdd30252542b6f8bf830326ac247cf9.png and $x4 = x0:

symmetries_of_particles_and_fields_e6b9d4d43f336a8cb37c7094d07c20eafd6d7164.png

which gives

symmetries_of_particles_and_fields_a0f8d2d9acb955222ed466e95b1804d224928467.png

Raising and lowering indices using the metric tensor

symmetries_of_particles_and_fields_7372f9980bede8138d7e05f47da362af950e73f5.png if and only if symmetries_of_particles_and_fields_cdbdfda567d42716e369cf1d2a5fb8bd61074f85.png is invertible.

Can use metric to raise and lower indices

symmetries_of_particles_and_fields_357636212a72a911c5f22e62ebf60282ce6a3a88.png

symmetries_of_particles_and_fields_b4f404e4ee72d852592e65b9902c8e13fb0ef978.png

Can raise indices of metric

symmetries_of_particles_and_fields_7db3afa9d92193b9526e1dfa450e5c803aea1f12.png

were we've contracted over symmetries_of_particles_and_fields_81c5f489718d4ff0ca6962103920c3133c0daea4.png. Then

symmetries_of_particles_and_fields_cd7f4971698f9623368a32374333e59a4c9c5683.png

which implies symmetries_of_particles_and_fields_3bae2e7beea0b2ffe4bb7161ba11d6634e20093d.png is the /inverse of symmetries_of_particles_and_fields_cdbdfda567d42716e369cf1d2a5fb8bd61074f85.png.

Same with Levi-Civita symbols

symmetries_of_particles_and_fields_f3e7514f4c30d657908b9602f9d71fcb82d4dc47.png

and derivatives wrt. $xμ

symmetries_of_particles_and_fields_9857cab4e957cdb6ebb3654b2d0b60e36b670d6a.png

If metric is definite

If metris is definite, the we can choose basis such that symmetries_of_particles_and_fields_9a93aeb93dc76279c61999804ee5fc97cf9e0a8b.png, i.e. can ignore distinction between upper and lower indices!

Matrix groups

Lorentz group

symmetries_of_particles_and_fields_eb7614e28e2f3a9cc59d35b17233e83d10117dc0.png

is called symmetries_of_particles_and_fields_10384a8302b1f940065116d95cff3136090fd994.png which has symmetries_of_particles_and_fields_73bd2a7d568217a44b74f6b4435c98371061a10a.png.

1D groups

All 1D Lie groups are such that the parameter symmetries_of_particles_and_fields_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png can be chosen such that

symmetries_of_particles_and_fields_d78fc41a10aa672266012f35ac960a9b92d1fd16.png

Furthermore, all 1D Lie groups are Abelian.

symmetries_of_particles_and_fields_c8dba34de599c574e6cb0c057a2c479101f3b46a.png and symmetries_of_particles_and_fields_b36faab4d1c324593f9962c6e9ef19ae85391384.png are isomorphic.

Let symmetries_of_particles_and_fields_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png be a symmetries_of_particles_and_fields_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png dimensional compact Abelian Lie group. Then

symmetries_of_particles_and_fields_6480a3712e774c5bea7ceadc653b2dd1767789a0.png

If symmetries_of_particles_and_fields_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is compact, then for every representation there exists an equivalent unitary representation.

Lie algebras

Generators

Consider group symmetries_of_particles_and_fields_a2306ebc6abb284ad5323ef88400cbcadb83b1d8.png where we choose symmetries_of_particles_and_fields_e36fc1c102e11d0610a06ae024e261bfbee72399.png with symmetries_of_particles_and_fields_89609db233f306341e26e8c53a349eb23eb12535.png such that

symmetries_of_particles_and_fields_4c4f2764fe925a1d84456ca7fad91dae6530a4ec.png

Taylors thm expand symmetries_of_particles_and_fields_66e41f2d19f308ba31bc2a90ccfd0f46400dcc68.png about symmetries_of_particles_and_fields_2d5a7877c4ac6d3cbfdae18a5f7ca2c0d079c93e.png, so we can write

symmetries_of_particles_and_fields_1f12eae023c81a4060cdaeabfc389f5236d2cb34.png

for "small" symmetries_of_particles_and_fields_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png.

Define

symmetries_of_particles_and_fields_4c92d8a8d251724d14555ccdf6b513af4de05607.png

Matrices symmetries_of_particles_and_fields_2ff58577dd3d98336afaa3238003cb226e01aa2c.png are the generators of the group.

The symmetries_of_particles_and_fields_b271f1c5e04249399969ceda30bb51429b96e0f1.png is put in to make symmetries_of_particles_and_fields_20b5ea1fe5bcd62a8e2e9c112ec912949390d393.png hermitian (in a unitary representation).

Examples

Example: symmetries_of_particles_and_fields_aad6e92ab821c571c6e39f5a75b806e13d278c4f.png

symmetries_of_particles_and_fields_5c22866f8ef6bf0e838a2a42208081283169410c.png

which implies

symmetries_of_particles_and_fields_b78e2c9991307fe6cfd5bf29a69780cdc4c3d6e8.png

Thus, the generators are hermitian and traceless.

Example: symmetries_of_particles_and_fields_64da3f5c2d1df788b3829f70ce11026c5689c9bc.png

We generally choose Pauli matrices as generators:

symmetries_of_particles_and_fields_75ed28d3db66f60ab7284a1d8bf485f6db1fbddb.png

Example: symmetries_of_particles_and_fields_fc9760dc0d1955b6154bd9d78fc2780922cf2046.png
  • Generators are Gell-Mann

    symmetries_of_particles_and_fields_318acea574e535f09ccf1e91ce37794e8bb1293a.png

Example: symmetries_of_particles_and_fields_17991b607bbb605ce7b8809fe5ec7099591778b8.png
  • Don't have constraint symmetries_of_particles_and_fields_2a631de00e15959f58e10bfc4bef5f0ca831f5e3.png.
  • Generators of symmetries_of_particles_and_fields_aad6e92ab821c571c6e39f5a75b806e13d278c4f.png, symmetries_of_particles_and_fields_2ff58577dd3d98336afaa3238003cb226e01aa2c.png are also the part a subset of the generators of symmetries_of_particles_and_fields_17991b607bbb605ce7b8809fe5ec7099591778b8.png, and "including the trace" we have symmetries_of_particles_and_fields_fa9b8d3ce56d4b0b4496c36a8f4f8592825deb69.png as the generators for symmetries_of_particles_and_fields_17991b607bbb605ce7b8809fe5ec7099591778b8.png
Example: symmetries_of_particles_and_fields_66610bc189a4a44d2732de6b79b5f5c39f742fa6.png

symmetries_of_particles_and_fields_7b37e10d5aad332b4c06e38a5155dd898ff7ffe1.png

for symmetries_of_particles_and_fields_66610bc189a4a44d2732de6b79b5f5c39f742fa6.png, symmetries_of_particles_and_fields_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is real, where we generally omit the symmetries_of_particles_and_fields_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png in the def. of generators.

Taking Taylor expansion, as for symmetries_of_particles_and_fields_aad6e92ab821c571c6e39f5a75b806e13d278c4f.png, we have

symmetries_of_particles_and_fields_f62ca6daf076eb712bc5b6286cb98f1212cded8a.png

i.e. symmetries_of_particles_and_fields_89a996c6a1e476834aafca19e023d0ff1378cb94.png are anti-symmetric.

Example: symmetries_of_particles_and_fields_b36faab4d1c324593f9962c6e9ef19ae85391384.png and symmetries_of_particles_and_fields_d00ec638831d2b816b9ba5942468c66ad49e4e85.png

For symmetries_of_particles_and_fields_b36faab4d1c324593f9962c6e9ef19ae85391384.png, generally choose:

symmetries_of_particles_and_fields_a874ee35348529de50935eb9c46164a1067a6a22.png

For symmetries_of_particles_and_fields_d00ec638831d2b816b9ba5942468c66ad49e4e85.png, generally choose

symmetries_of_particles_and_fields_528b67c8e92aeefb301907f69ec8eca4f03f8896.png

Additional constraint symmetries_of_particles_and_fields_922427797210037b47593a207429011dfc7b884a.png is automatically satisfied (why?).

Commutations

Consider symmetries_of_particles_and_fields_d8a5fce66989dc5c0058e3c0fa5aa3143b6ed2c4.png with symmetries_of_particles_and_fields_507fc22db7dfdc0f8ca2965fb03d32d52f6783ca.png.

Expanding each of these using their Taylor expansion up to and including 2nd order terms, then take the commutation relation between symmetries_of_particles_and_fields_ec0a34e0a656018252ed6628c2d899d959753781.png and symmetries_of_particles_and_fields_9e121140127753df34d3cf56c2e2fee5af372233.png, the expansion of symmetries_of_particles_and_fields_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png and symmetries_of_particles_and_fields_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png, respectively, we have

symmetries_of_particles_and_fields_930bc0f4b7604ee5199d8a3a00d58df07119181b.png

For this to be true symmetries_of_particles_and_fields_aa8d1e2e6b415d959cbb35afd2b6038851854982.png we must have

symmetries_of_particles_and_fields_07a50415d2637512452658a396cab7ef03297524.png

for some numbers symmetries_of_particles_and_fields_023a97b89abf5a4193d931694487b40ce7677c95.png.

Which implies

symmetries_of_particles_and_fields_6ff84a611e0adc801b93ffd7a88180d4ee271cfd.png

where symmetries_of_particles_and_fields_081012e045dffd171803412b9fd84ee6daa69ff1.png are called the structure constants.

Exponentiation

Consider 1D subgroup symmetries_of_particles_and_fields_71b0c4c919d23064e2314a158913f54e6052ac11.png, symmetries_of_particles_and_fields_0133d902de9f6d05d35249bc1e78fcf8e638cedb.png labeled by the single parameter symmetries_of_particles_and_fields_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png.

Chose symmetries_of_particles_and_fields_570462f792cbe96a48effcfae87b005cefc599cb.png such that

symmetries_of_particles_and_fields_d91164561b9a267c108079da78ebc4e813e06596.png

Diff. wrt. symmetries_of_particles_and_fields_aa1698cb8ee1665238ec3e91824191643c62ee93.png

symmetries_of_particles_and_fields_bf9b7174bfd0ffc4367ce51b6839967a47b26e86.png

and set symmetries_of_particles_and_fields_5dc22e2cf6500b939cead3609b2a7dcd765c45bb.png, which gives us a differential equation for all symmetries_of_particles_and_fields_9f5048834ae702cb1a8486235e8105df13f0990f.png:

symmetries_of_particles_and_fields_04f9f815632850dca28344ed080536a6e0318ec3.png

which has the solution

symmetries_of_particles_and_fields_a9a1defbf162ebbac0a330bec076f9080fea13ee.png

Or, if you consider this in the familiar notation used in Diff. Geom. for the exponential map, we can write this as

symmetries_of_particles_and_fields_8fa21469c41cb3aa7f75cb959b7c7b7306a85419.png

where symmetries_of_particles_and_fields_c72a32d6011b117fc2070f12735577031fe8a506.png, s.t.

symmetries_of_particles_and_fields_36c440478150ebb39bfc80f0e0879198d870238e.png

where symmetries_of_particles_and_fields_c0407eba5c5ae5efed6c8209e18c4a615e520d9e.png, i.e. maps from the Lie algebra to the Lie group.

Going back to the earier notation of symmetries_of_particles_and_fields_9f5048834ae702cb1a8486235e8105df13f0990f.png, we can write

symmetries_of_particles_and_fields_6695a51bcfb06a02d30019cca1c03a584ea73290.png

where

symmetries_of_particles_and_fields_5f520a6292ca69b3da0787c6b53bc48e1e3baf26.png

which implies

symmetries_of_particles_and_fields_5bc42759fcd74541914a09a929f1a5de83092064.png

Adjoint representation

Can define an adjoint representation:

symmetries_of_particles_and_fields_a4d85ebb551d3093f54918b90a7ef698f0769bb0.png

then

symmetries_of_particles_and_fields_94ad8a1b7d70db7d9215cf566cec0dea5dbd5c33.png

Examples
  • symmetries_of_particles_and_fields_64da3f5c2d1df788b3829f70ce11026c5689c9bc.png and symmetries_of_particles_and_fields_d00ec638831d2b816b9ba5942468c66ad49e4e85.png
    • symmetries_of_particles_and_fields_64da3f5c2d1df788b3829f70ce11026c5689c9bc.png take generators symmetries_of_particles_and_fields_84a6c1c4f9ce1e317d64dbd307efdc393e0b8248.png, with

      symmetries_of_particles_and_fields_243b8ae7cfa157a2959ba7fa0db0e21f360ea2a0.png

      and so the structure constants are symmetries_of_particles_and_fields_fd1f3c5be2cb9ab4aafe1dcbafaf790b7a3f3c2e.png.

    • Consider

      symmetries_of_particles_and_fields_6b3c848daa1a6c7d661c15db15dce6d562fa09ae.png

      Then

      symmetries_of_particles_and_fields_3239d66df2110f1cec252f63a4e9ef98327364b4.png

      So if we write symmetries_of_particles_and_fields_ec409b6f6e16e3b907bcf9e1549a9fba6296344e.png, then we can write

      symmetries_of_particles_and_fields_900614407d4e3c503539702543f1779ce24c8f06.png

      which is just rotation in symmetries_of_particles_and_fields_492d525117d0dcc93d066c8759f46b98cf9980ca.png!

    • Therefore the adjoint representation of symmetries_of_particles_and_fields_64da3f5c2d1df788b3829f70ce11026c5689c9bc.png is the defining or fundamental representation of symmetries_of_particles_and_fields_d00ec638831d2b816b9ba5942468c66ad49e4e85.png. hence

      symmetries_of_particles_and_fields_0cee5bb13d4c2ab4ecfe42eafe85144d943c56a5.png

      for elements "close to the identity".

    In general, the adjoint representation is not the fundamental representation of some other group…

Killing form

The Killing form

symmetries_of_particles_and_fields_206d0b4ca263b84d2b9efcd6ac335970e87ff2b3.png

real symmetric symmetries_of_particles_and_fields_b8d9432836c408cbe101834165621d097c93f7ec.png matrix.

Usually we'll choose diagonal basis such that

symmetries_of_particles_and_fields_df286494f677f61d0a8a77f363aeb23f8626b8a0.png

Examples

symmetries_of_particles_and_fields_64da3f5c2d1df788b3829f70ce11026c5689c9bc.png

symmetries_of_particles_and_fields_89df499cd8d5e760a29335f43dd24f3c92c8cba8.png

So we rescale generators such that

symmetries_of_particles_and_fields_eb758e6f4aa92cf37205c05f07409984fb551a1b.png

Application of killing form

  • Can use Killing form to lower and raise indices
  • E.g. kan define structure consts with all lower indices

    symmetries_of_particles_and_fields_198df6ddcab690542283a40ebfa4ccf4bd49abe7.png

    • This is an invariant tensor → can be written as a product of generators

      symmetries_of_particles_and_fields_5d9fdbf39e807bcdb6f1cf078fc24b695bf3c3b0.png

Invariant Tensors

Let symmetries_of_particles_and_fields_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png be a Lie group.

An invariant tensor

symmetries_of_particles_and_fields_574b60bcee3208399ee37283ad28cce86842988f.png

is a tensor that remains invariant under the actions of group symmetries_of_particles_and_fields_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, i.e.

symmetries_of_particles_and_fields_553870c2a9279778992a327120b48085d05a84ff.png