From 8ec91ae4dac57f35176488b2b913378388285276 Mon Sep 17 00:00:00 2001 From: Ramunas Girdziusas Date: Thu, 26 Sep 2024 20:29:32 +0300 Subject: [PATCH] minor edit --- README.md | 10 ++++++---- index.html | 18 +++++++++--------- index.md | 18 +++++++++--------- 3 files changed, 24 insertions(+), 22 deletions(-) diff --git a/README.md b/README.md index 3e48739..01d73da 100644 --- a/README.md +++ b/README.md @@ -1,15 +1,17 @@ # Introduction -This are my notes taken while reading [Shankland (1970)](https://aapt.scitation.org/doi/10.1119/1.1976018). +These are my notes taken while reading [Shankland (1970)](https://aapt.scitation.org/doi/10.1119/1.1976018). -Keywords: theoretical physics, spinors, irreducible representations, tensor fields, tensor algebras, spin, Faddeev, LeVerrier. +Keywords: theoretical physics, tensor field, tensor algebra, spin, spinor, Weinberg. # Setup -Clone the repo and run +To reuse LaTeX and gomarkdown, clone the repo and run ```bash ./gocode/md2html ``` -if you want to reuse LaTeX and gomarkdown. +Deploy with [Github Pages](https://medium.com/flycode/how-to-deploy-a-static-website-for-free-using-github-pages-8eddc194853b). + + diff --git a/index.html b/index.html index 79d9f01..fef9e78 100644 --- a/index.html +++ b/index.html @@ -158,7 +158,7 @@

The Spectrum of a Tensor Field

  • \((\frac{1}{2},0)\oplus (0,\frac{1}{2})\): A full single spinor index. Shankland’s doublet and its antidoublet: \(\frac{1}{2}, \frac{1}{2}\) subspaces with multiplicites \(2\) and \(2\).

    • -

  • \((1,1)\): Two symmetric tensor indices. A mismatch with Shankland’s pentuplet, triplet, and two singlets: Subspaces \(0, 1, 2\) with the multiplicities \(1\), \(3\), and \(5\). Where is the missing singlet? In a symmetric two-index tensor case, to remove a singlet also means to make the tensor traceless, so the group theory still matches Shankland under the assumption of tracelessness.

    • +

  • \((1,1)\): Two symmetric tensor indices. A mismatch with Shankland’s pentuplet, triplet, and two singlets: Subspaces \(0, 1, 2\) with multiplicities \(1\), \(3\), and \(5\). Where is the missing singlet? In a symmetric two-index tensor case, to remove a singlet also means to make the tensor traceless, so the group theory still matches Shankland under the assumption of tracelessness.

  • \((1,0)\oplus (0,1)\): Two asymmetric tensor indices. Shankland’s two particle triplets: Subspaces \(1\) and \(1\) with the multiplicities \(3\) and \(3\).

    • @@ -173,25 +173,25 @@

    More Tensor Charades

    One can find some other mildly successful uses/hints of tensor algebras in Phys. Rev. 106, 1345 (1957); Nuovo Cimento, 43, 475 (1966); Nuovo Cimento 47, 145 (1967); Phys. Rev. 153, 1652 (1967); Phys. Rev. 161, 1631 (1967); Phys. Rev. D 8, 2650 (1973); Nuovo Cimento 28, 409 (1975); Phys. Lett. B 301 4 339 (1993); Phys. Rev. C 64, 015203 (2001); Phys. Rev. D 64, 125013 (2001); Hadronic J. 26, 351 (2003); Phys. Rev. D 67, 085021 (2003); Phys. Rev. D 67, 125011 (2003); Nucl. Phys. B724, 453 (2005); Phys. Rev. D 74, 084036 (2006); P. Cvitanović (2008); V. Monchiet and G. Bonnet (2010); Phys. Rev. D 97, 115043 (2018); SUGRA and CDC…

    -

    It is tough to read this literature, nothing is too interesting there, to be honest. Note that Shankland’s paper is hardly known. It is not in Phys Rev spaces, the AAPT community has not dissected it inside out like it would do with anything touching Maxwell.

    +

    It is tough to read this literature, and the results seldom justify the complexity. Note that Shankland’s paper is hardly known. It is not in Phys Rev spaces, the AAPT community has not dissected it inside out like it would do with anything touching Maxwell.

    Is Shankland’s Program Worthy?

    -

    We do not get theorems or new results, only a preliminary space to build stuff. There is no dynamics yet, but this is a world in its own. I am thinking of Apocalypto (2006), esp. its ending which is also the new beginning.

    +

    We do not get theorems or new results, only “a kinematic space”, there is no dynamics yet. I am thinking of Apocalypto (2006), esp. its ending which is also the new beginning.

    -

    As a concrete example, Shankland shows how removing spin 0 from a vector field leads to apesanteur \(A\) aka a vector potential. We can now apply this to gravity by removing spins 0 and 1 from the symmetric tensor field and get some kind of an apesanteur there too. Honestly, I would do this only if someone pressed a gun to my brain.

    +

    As a concrete example, Shankland shows how removing spin 0 from a vector field leads to apesanteur \(A\) aka a vector potential. We can now apply this to gravity by removing spins 0 and 1 from the symmetric tensor field and get some kind of an apesanteur there too. Maybe even rederive SUGRA projectors. Honestly, I would do this only if someone pressed a gun to my brain.

    -

    So a field with tensor/spinor indices and their permutation symmetries in, a Lorentz-invariant operator to build a quadratic form for the field out. With a machinery to engineer (add remove) spin content. Notice that Shankland calls a quadratic form operator “a correlation function” which is probably another hidden use, albeit little explored.

    +

    So a field with tensor/spinor indices and their permutation symmetries in, a Lorentz-invariant operator to build a quadratic form for the field out. With a machinery to engineer (add remove) spin content. Notice that Shankland calls a quadratic form operator “a correlation function” which is probably another use (QFT), albeit little explored.

    -

    A lot of technical questions are left unanswered. The gauge transforms are barely discussed, if at all. Also when and why do the combinations of the primitives such as \(k\), \(g\), \(\gamma\) would form an algebra? Where do these primitives come from for an arbitrary Lie group/algebra? Have we not missed the Pauli matrix or \(\epsilon\) here and there? How to complete an algebra, verify the basis dimension? Why does \(\gamma_{\mu}p^{\mu}\) have to be treated like an independent quantity when building a vector-spinor basis, effectively doubling the basis dimension from 5 to 10?

    +

    A lot of technical questions are left unanswered. The gauge transforms are barely discussed. When and why do the combinations of the primitives such as \(k\), \(g\), \(\gamma\) would form an algebra? Where do these primitives come from for an arbitrary Lie group/algebra? Have we not missed the Pauli matrix or the Levi Civita symbol \(\epsilon\)? How to complete an algebra, verify the basis dimension? Why is \(\gamma_{\mu}p^{\mu}\) treated like an independent quantity when building a vector-spinor basis, effectively doubling the basis dimension from 5 to 10?

    -

    A fundamental problem here is that a desire to have a quadratic form/matrix/correlation doubles all the indices, but this complexity might be avoidable. A worthy contender “engine” would be what one would call the “Lie-Rivlin-Spencer-Zhilin” pipeline, though it is only in my head at the moment. It would be yet another rather long program to build invariant cost functions, by using the same primitives of Shankland, but more frugally, by not relying on the quadratic form operator, going only for the cost function directly. It would be harder to engineer spin content though.

    +

    In addition to magic, a fundamental problem here is that a desire to have a quadratic form/matrix/correlation doubles all the indices, but this complexity might be avoidable. A worthy contender “engine” would be what one would call the “Lie-Rivlin-Spencer-Zhilin” theory, though this is only in my head at the moment. It would be yet another rather long program to build invariant cost functions, by using the same primitives of Shankland, but more frugally. By not relying on the quadratic form operator, going only for the cost function, directly. It would be harder to engineer spin content though.

    How Not to Get Lost in Modern Physics

    -

    I wish I knew. The theory revolves around the Lorentz, PCT, and gauge transformations. This does not sound much until you look into Weinberg’s three volumes of QFT, and his two volumes of classical gravity.

    +

    I wish I knew. The theory revolves around the Lorentz, PCT, and gauge transformations. This does not sound much until one encounters Weinberg’s three volumes of QFT, and his two volumes of classical gravity.

    -

    History and real experiments compress it all naturally. We do not have that many key experiments after 400 years since Galileo. Consider light: Newton, Fresnel-Arago, Hertz, Lebedev, Compton, Pound–Rebka, Breit–Wheeler… Add a dozen more, but the subject will remain manageable.

    +

    History and real experiments compress it all naturally. We do not have that many key experiments after 400 years since Galileo. Consider light: Newton, Fresnel-Arago, Hertz, Lebedev, Compton, Breit–Wheeler, Pound–Rebka… Add a dozen more, the subject should remain manageable.

    Sign the contract big boy... diff --git a/index.md b/index.md index 197e480..a70b73d 100644 --- a/index.md +++ b/index.md @@ -155,7 +155,7 @@ According to group theory, combining indices means taking "tensor products $(m,n - $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$: A full single spinor index. Shankland's doublet and its antidoublet: $\frac{1}{2}, \frac{1}{2}$ subspaces with multiplicites $2$ and $2$. -- $(1,1)$: Two symmetric tensor indices. A mismatch with Shankland's pentuplet, triplet, and two singlets: Subspaces $0, 1, 2$ with the multiplicities $1$, $3$, and $5$. Where is the missing singlet? In a symmetric two-index tensor case, to remove a singlet also means to make the tensor traceless, so the group theory still matches Shankland under the assumption of **tracelessness**. +- $(1,1)$: Two symmetric tensor indices. A mismatch with Shankland's pentuplet, triplet, and two singlets: Subspaces $0, 1, 2$ with multiplicities $1$, $3$, and $5$. Where is the missing singlet? In a symmetric two-index tensor case, to remove a singlet also means to make the tensor traceless, so the group theory still matches Shankland under the assumption of **tracelessness**. - $(1,0)\oplus (0,1)$: Two asymmetric tensor indices. Shankland's two particle triplets: Subspaces $1$ and $1$ with the multiplicities $3$ and $3$. @@ -169,25 +169,25 @@ Shankland's construction is one of the most complex calculations that one can bu One can find some other mildly successful uses/hints of tensor algebras in [Phys. Rev. 106, 1345 (1957)](https://journals.aps.org/pr/abstract/10.1103/PhysRev.106.1345); [Nuovo Cimento, 43, 475 (1966)](https://link.springer.com/article/10.1007/BF02752873); [Nuovo Cimento 47, 145 (1967)](https://link.springer.com/article/10.1007/BF02818340); [Phys. Rev. 153, 1652 (1967)](https://journals.aps.org/pr/abstract/10.1103/PhysRev.153.1652); [Phys. Rev. 161, 1631 (1967)](https://journals.aps.org/pr/abstract/10.1103/PhysRev.161.1631); [Phys. Rev. D 8, 2650 (1973)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.8.2650); [Nuovo Cimento 28, 409 (1975)](https://inspirehep.net/literature/98459); [Phys. Lett. B 301 4 339 (1993)](https://arxiv.org/abs/hep-th/9212008); [Phys. Rev. C 64, 015203 (2001)](https://arxiv.org/abs/hep-ph/0103172); [Phys. Rev. D 64, 125013 (2001)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.64.125013); [Hadronic J. 26, 351 (2003)](https://www.imath.kiev.ua/~nikitin/PAPER26.pdf); [Phys. Rev. D 67, 085021 (2003)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.67.085021); [Phys. Rev. D 67, 125011 (2003)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.67.125011); [Nucl. Phys. B724, 453 (2005)](https://arxiv.org/abs/hep-th/0505255); [Phys. Rev. D 74, 084036 (2006)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.74.084036); [P. Cvitanović (2008)](https://birdtracks.eu/); [V. Monchiet and G. Bonnet (2010)](https://royalsocietypublishing.org/doi/10.1098/rspa.2010.0149); [Phys. Rev. D 97, 115043 (2018)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.97.115043); [SUGRA and CDC](https://news.stonybrook.edu/facultystaff/qa-with-breakthrough-prize-winner-peter-van-nieuwenhuizen/)... -It is tough to read this literature, nothing is too interesting there, to be honest. Note that Shankland's paper is hardly known. It is not in Phys Rev spaces, the AAPT community has not dissected it inside out like it would do with anything touching Maxwell. +It is tough to read this literature, and the results seldom justify the complexity. Note that Shankland's paper is hardly known. It is not in Phys Rev spaces, the AAPT community has not dissected it inside out like it would do with anything touching Maxwell. ## Is Shankland's Program Worthy? -We do not get theorems or new results, only a preliminary space to build stuff. There is no dynamics yet, but this is a world in its own. I am thinking of [Apocalypto (2006)](https://www.imdb.com/title/tt0472043/), esp. its ending which is also the new beginning. +We do not get theorems or new results, only "a kinematic space", there is no dynamics yet. I am thinking of [Apocalypto (2006)](https://www.imdb.com/title/tt0472043/), esp. its ending which is also the new beginning. -As a concrete example, Shankland shows how removing spin 0 from a vector field leads to apesanteur $A$ aka a vector potential. We can now apply this to gravity by removing spins 0 and 1 from the symmetric tensor field and get some kind of an apesanteur there too. Honestly, I would do this only if someone pressed a gun to my brain. +As a concrete example, Shankland shows how removing spin 0 from a vector field leads to apesanteur $A$ aka a vector potential. We can now apply this to gravity by removing spins 0 and 1 from the symmetric tensor field and get some kind of an apesanteur there too. Maybe even rederive SUGRA projectors. Honestly, I would do this only if someone pressed a gun to my brain. -So a field with tensor/spinor indices and their permutation symmetries in, a Lorentz-invariant operator to build a quadratic form for the field out. With a machinery to engineer (add remove) spin content. Notice that Shankland calls a quadratic form operator "a correlation function" which is probably another hidden use, albeit little explored. +So a field with tensor/spinor indices and their permutation symmetries in, a Lorentz-invariant operator to build a quadratic form for the field out. With a machinery to engineer (add remove) spin content. Notice that Shankland calls a quadratic form operator "a correlation function" which is probably another use (QFT), albeit little explored. -A lot of technical questions are left unanswered. The gauge transforms are barely discussed, if at all. Also when and why do the combinations of the primitives such as $k$, $g$, $\gamma$ would form an algebra? Where do these primitives come from for an arbitrary Lie group/algebra? Have we not missed the Pauli matrix or [$\epsilon$](https://en.wikipedia.org/wiki/Levi-Civita_symbol) here and there? How to complete an algebra, verify the basis dimension? Why does $\gamma_{\mu}p^{\mu}$ have to be treated like an independent quantity when building a vector-spinor basis, effectively doubling the basis dimension from 5 to 10? +A lot of technical questions are left unanswered. The gauge transforms are barely discussed. When and why do the combinations of the primitives such as $k$, $g$, $\gamma$ would form an algebra? Where do these primitives come from for an arbitrary Lie group/algebra? Have we not missed the Pauli matrix or the Levi Civita symbol [$\epsilon$](https://en.wikipedia.org/wiki/Levi-Civita_symbol)? How to complete an algebra, verify the basis dimension? Why is $\gamma_{\mu}p^{\mu}$ treated like an independent quantity when building a vector-spinor basis, effectively doubling the basis dimension from 5 to 10? -A fundamental problem here is that a desire to have a quadratic form/matrix/correlation doubles all the indices, but this complexity might be avoidable. A worthy contender "engine" would be what one would call the "Lie-Rivlin-Spencer-Zhilin" pipeline, though it is only in my head at the moment. It would be yet another rather long program to build invariant cost functions, by using the same primitives of Shankland, but more frugally, by not relying on the quadratic form operator, going only for the cost function directly. It would be harder to engineer spin content though. +In addition to magic, a fundamental problem here is that a desire to have a quadratic form/matrix/correlation doubles all the indices, but this complexity might be avoidable. A worthy contender "engine" would be what one would call the "Lie-Rivlin-Spencer-Zhilin" theory, though this is only in my head at the moment. It would be yet another rather long program to build invariant cost functions, by using the same primitives of Shankland, but more frugally. By not relying on the quadratic form operator, going only for the cost function, directly. It would be harder to engineer spin content though. ## How Not to Get Lost in Modern Physics -I wish I knew. The theory revolves around the Lorentz, PCT, and gauge transformations. This does not sound much until you look into Weinberg's three volumes of QFT, and his two volumes of classical gravity. +I wish I knew. The theory revolves around the Lorentz, PCT, and gauge transformations. This does not sound much until one encounters Weinberg's three volumes of QFT, and his two volumes of classical gravity. -History and real experiments compress it all naturally. We do not have that many key experiments after 400 years since Galileo. Consider light: Newton, Fresnel-Arago, Hertz, Lebedev, Compton, [Pound–Rebka](https://en.wikipedia.org/wiki/Pound%E2%80%93Rebka_experiment), [Breit–Wheeler](https://en.wikipedia.org/wiki/Breit%E2%80%93Wheeler_process)... Add a dozen more, but the subject will remain manageable. +History and real experiments compress it all naturally. We do not have that many key experiments after 400 years since Galileo. Consider light: Newton, Fresnel-Arago, Hertz, Lebedev, Compton, [Breit–Wheeler](https://en.wikipedia.org/wiki/Breit%E2%80%93Wheeler_process), [Pound–Rebka](https://en.wikipedia.org/wiki/Pound%E2%80%93Rebka_experiment)... Add a dozen more, the subject should remain manageable.
    Sign the contract big boy...