How to represent a conformal transformation using spinors?
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In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan space with metric as being of the from:
$$g_{munu}longrightarrowlambda^{2}g_{munu}$$
Where $lambda$ may be a function of the coordinates. In general I understand you can represent a conformal transformation as a composition of this dilation (our $lambda)$ with a rotation (lets call that $R$) which in component form looks like:
$$tilde{g}_{alphabeta}=lambda R_{alpha}^{mu}g_{munu}R_{beta}^{nu}lambda=lambda R^{T}gRlambda$$
Where the latter term is written in matrix form. So how do we get from here to a spinor representation of our transformation??? Note I'm just looking at conformal transformations of $R^{3,1}$ (with a point at infinity to compactify it).
I'm thinking something like:
$$tilde{g}_{munu}=bar{s}g_{munu}s$$
Where $s$ is the spinor and the bar denotes it's complex conjugate
but I'm not sure that it actually has the required properties.
metric-spaces conformal-geometry
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
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add a comment |
$begingroup$
In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan space with metric as being of the from:
$$g_{munu}longrightarrowlambda^{2}g_{munu}$$
Where $lambda$ may be a function of the coordinates. In general I understand you can represent a conformal transformation as a composition of this dilation (our $lambda)$ with a rotation (lets call that $R$) which in component form looks like:
$$tilde{g}_{alphabeta}=lambda R_{alpha}^{mu}g_{munu}R_{beta}^{nu}lambda=lambda R^{T}gRlambda$$
Where the latter term is written in matrix form. So how do we get from here to a spinor representation of our transformation??? Note I'm just looking at conformal transformations of $R^{3,1}$ (with a point at infinity to compactify it).
I'm thinking something like:
$$tilde{g}_{munu}=bar{s}g_{munu}s$$
Where $s$ is the spinor and the bar denotes it's complex conjugate
but I'm not sure that it actually has the required properties.
metric-spaces conformal-geometry
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
$endgroup$
1
$begingroup$
You might find some useful hints in arxiv.org/pdf/1101.2334.pdf arxiv.org/pdf/1506.01438.pdf and dwc.knaw.nl/DL/publications/PU00017567.pdf
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– iSeeker
Dec 9 '18 at 18:15
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@iSeeker Thank you the first paper goes along the lines of why I asked my question, (but I'd never seen it). The second I've read, but still holds value (:
$endgroup$
– R. Rankin
Dec 9 '18 at 23:04
add a comment |
$begingroup$
In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan space with metric as being of the from:
$$g_{munu}longrightarrowlambda^{2}g_{munu}$$
Where $lambda$ may be a function of the coordinates. In general I understand you can represent a conformal transformation as a composition of this dilation (our $lambda)$ with a rotation (lets call that $R$) which in component form looks like:
$$tilde{g}_{alphabeta}=lambda R_{alpha}^{mu}g_{munu}R_{beta}^{nu}lambda=lambda R^{T}gRlambda$$
Where the latter term is written in matrix form. So how do we get from here to a spinor representation of our transformation??? Note I'm just looking at conformal transformations of $R^{3,1}$ (with a point at infinity to compactify it).
I'm thinking something like:
$$tilde{g}_{munu}=bar{s}g_{munu}s$$
Where $s$ is the spinor and the bar denotes it's complex conjugate
but I'm not sure that it actually has the required properties.
metric-spaces conformal-geometry
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
$endgroup$
In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan space with metric as being of the from:
$$g_{munu}longrightarrowlambda^{2}g_{munu}$$
Where $lambda$ may be a function of the coordinates. In general I understand you can represent a conformal transformation as a composition of this dilation (our $lambda)$ with a rotation (lets call that $R$) which in component form looks like:
$$tilde{g}_{alphabeta}=lambda R_{alpha}^{mu}g_{munu}R_{beta}^{nu}lambda=lambda R^{T}gRlambda$$
Where the latter term is written in matrix form. So how do we get from here to a spinor representation of our transformation??? Note I'm just looking at conformal transformations of $R^{3,1}$ (with a point at infinity to compactify it).
I'm thinking something like:
$$tilde{g}_{munu}=bar{s}g_{munu}s$$
Where $s$ is the spinor and the bar denotes it's complex conjugate
but I'm not sure that it actually has the required properties.
metric-spaces conformal-geometry
metric-spaces conformal-geometry
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
edited Dec 2 '18 at 9:16
R. Rankin
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
asked Dec 2 '18 at 9:05
R. RankinR. Rankin
333213
333213
1
$begingroup$
You might find some useful hints in arxiv.org/pdf/1101.2334.pdf arxiv.org/pdf/1506.01438.pdf and dwc.knaw.nl/DL/publications/PU00017567.pdf
$endgroup$
– iSeeker
Dec 9 '18 at 18:15
$begingroup$
@iSeeker Thank you the first paper goes along the lines of why I asked my question, (but I'd never seen it). The second I've read, but still holds value (:
$endgroup$
– R. Rankin
Dec 9 '18 at 23:04
add a comment |
1
$begingroup$
You might find some useful hints in arxiv.org/pdf/1101.2334.pdf arxiv.org/pdf/1506.01438.pdf and dwc.knaw.nl/DL/publications/PU00017567.pdf
$endgroup$
– iSeeker
Dec 9 '18 at 18:15
$begingroup$
@iSeeker Thank you the first paper goes along the lines of why I asked my question, (but I'd never seen it). The second I've read, but still holds value (:
$endgroup$
– R. Rankin
Dec 9 '18 at 23:04
1
1
$begingroup$
You might find some useful hints in arxiv.org/pdf/1101.2334.pdf arxiv.org/pdf/1506.01438.pdf and dwc.knaw.nl/DL/publications/PU00017567.pdf
$endgroup$
– iSeeker
Dec 9 '18 at 18:15
$begingroup$
You might find some useful hints in arxiv.org/pdf/1101.2334.pdf arxiv.org/pdf/1506.01438.pdf and dwc.knaw.nl/DL/publications/PU00017567.pdf
$endgroup$
– iSeeker
Dec 9 '18 at 18:15
$begingroup$
@iSeeker Thank you the first paper goes along the lines of why I asked my question, (but I'd never seen it). The second I've read, but still holds value (:
$endgroup$
– R. Rankin
Dec 9 '18 at 23:04
$begingroup$
@iSeeker Thank you the first paper goes along the lines of why I asked my question, (but I'd never seen it). The second I've read, but still holds value (:
$endgroup$
– R. Rankin
Dec 9 '18 at 23:04
add a comment |
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$begingroup$
You might find some useful hints in arxiv.org/pdf/1101.2334.pdf arxiv.org/pdf/1506.01438.pdf and dwc.knaw.nl/DL/publications/PU00017567.pdf
$endgroup$
– iSeeker
Dec 9 '18 at 18:15
$begingroup$
@iSeeker Thank you the first paper goes along the lines of why I asked my question, (but I'd never seen it). The second I've read, but still holds value (:
$endgroup$
– R. Rankin
Dec 9 '18 at 23:04