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==Examples== |
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==Examples== |
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<ol> |
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1. <math>-i\partial_x</math> is a Dirac operator on the tangent bundle over a line. |
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<li><math>-i\partial_x</math> is a Dirac operator on the tangent bundle over a line. </li><br /> |
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2: We now consider a simple bundle of importance in physics: The configuration space of a particle |
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with spin <sup>1</sup>⁄<sub>2</sub> confined to a plane, which is also the base manifold. It's represented by a wavefunction ψ: '''R'''<sup>2</sup> → '''C'''<sup>2</sup> |
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:<math>\begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix} ,</math> |
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where ''x'' and ''y'' are the usual coordinate functions on '''R'''<sup>2</sup>. |
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χ specifies the [[probability amplitude]] for the particle to be in the |
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spin-up state, and similarly for η. The so-called [[spin-Dirac operator]] |
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can then be written |
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<li> We now consider a simple bundle of importance in physics: The configuration space of a particle with spin <sup>1</sup>⁄<sub>2</sub> confined to a plane, which is also the base manifold. It's represented by a a wavefunction ψ: '''R'''<sup>2</sup> → '''C'''<sup>2</sup> |
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:<math>\begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix}</math >,<br /> |
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where ''x'' and ''y'' are the usual coordinate functions on '''R'''<sup>2</sup>. χ specifies the [[probability amplitude]] for the particle to be in the spin-up state, and similarly for η. The so-called [[spin-Dirac operator]] can then be written |
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:<math>D=-i\sigma_x\partial_x-i\sigma_y\partial_y,\,</math> |
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:<math>D=-i\sigma_x\partial_x-i\sigma_y\partial_y,\,</math> |
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where σ<sub>''i''</sub> are the [[Pauli matrices]]. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a [[Clifford algebra]]. |
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where σ<sub>''i''</sub> are the [[Pauli matrices]]. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a [[Clifford algebra]]. |
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Solutions to the Dirac equation for spinor fields are often called ''harmonic spinors''[http://eom.springer.de/S/s086780.htm]. |
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Solutions to the Dirac equation for spinor fields are often called ''harmonic spinors''[http://eom.springer.de/S/s086780.htm].</li><br /> |
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3: The most famous Dirac operator describes the propagation of a free [[fermion]] in |
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three dimensions and is elegantly written |
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<li> The most famous Dirac operator describes the propagation of a free [[fermion]] in three dimensions and is elegantly written |
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:<math>D=\gamma^\mu\partial_\mu\ \equiv \partial\!\!\!/,</math> |
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:<math>D=\gamma^\mu\partial_\mu\ \equiv \partial\!\!\!/,</math> |
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using the [[Feynman slash notation]]. </li><br /> |
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using the [[Feynman slash notation]]. |
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4: There is also the Dirac operator arising in [[Clifford analysis]]. In euclidean n-space this is |
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:<math>D=\ Sigma_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}</math> |
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<li> There is also the Dirac operator arising in [[Clifford analysis]]. In euclidean n-space this is |
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:<math>D=\ sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}</math> |
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where |
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where |
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:<math> \{e_{j}:j=1,\ldots, n\}</math> |
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:<math> \{e_{j}:j=1,\ldots, n\}</math> |
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is an orthonormal basis for euclidean n-space, and <math>\mathbb{R}^{n}</math> is considered to be embedded in a [[Clifford algebra]]. |
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This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a [[spinor bundle]]. </li><br /> |
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is an orthonormal basis for euclidean n-space, and <math>\mathbb{R}^{n}</math> is considered to be embedded in a [[Clifford algebra]]. |
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<li> For a [[spin manifold]], M, the Atiyah-Singer-Dirac operator is locally defined as follows: For <math>x\in M</math> and <math>e_{1}(x),\ldots,e_{j}(x)</math> a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is |
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This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a [[spinor bundle]]. |
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:<math>\sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}</math>, |
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where <math>\tilde{\Gamma}</math> is a lifting of the Levi-Civita connection on M to the [[spinor bundle]] over M.</li> |
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</ol> |
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5: For a [[spin manifold]], M, the Atiyah-Singer-Dirac operator is locally defined as follows: For <math>x\in M</math> and <math>e_{1}(x),\ldots,e_{j}(x)</math> a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is <math>\Sigma_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}</math>, where <math>\tilde{\Gamma}</math> is a lifting of the Levi-Civita connection on M to the [[spinor bundle]] over M. |
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==See also== |
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==See also== |
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors.
In general, let
be a first-order differential operator acting on a
vector bundle
over a Riemannian manifold
.
If
![{\displaystyle D^{2}=\Delta ,\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaNw1yjmQzAoPa2i4zAo5zAw3yje2otnDotnDaDdAote5nAdEntC1)
with
being the Laplacian of
,
is called a Dirac operator.
In high-energy physics, this requirement is often relaxed: only the second-order part
of
must equal the Laplacian.
Examples
is a Dirac operator on the tangent bundle over a line.
- We now consider a simple bundle of importance in physics: The configuration space of a particle with spin 1⁄2 confined to a plane, which is also the base manifold. It's represented by a a wavefunction ψ: R2 → C2
,
where x and y are the usual coordinate functions on R2. χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written
![{\displaystyle D=-i\sigma _{x}\partial _{x}-i\sigma _{y}\partial _{y},\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zNK0yjs3aqvEagrFz2nBnqiNaDw0oqzEaAiQaAdFoDBFzDiPyjsO)
where σi are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a Clifford algebra.
Solutions to the Dirac equation for spinor fields are often called harmonic spinors[1].
- The most famous Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written
![{\displaystyle D=\gamma ^{\mu }\partial _{\mu }\ \equiv \partial \!\!\!/,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85a2dEo2dEaNKQygw3o2e1ots3nthBztm2otC4nAvAaNC4yga3nDlE)
using the Feynman slash notation.
- There is also the Dirac operator arising in Clifford analysis. In euclidean n-space this is
![{\displaystyle D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CaDm3zNG3z2s4zjKNyga0zDs1yja0zja5aDzEzNdCoDFDaNvFago1)
where
![{\displaystyle \{e_{j}:j=1,\ldots ,n\}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zNdBotnAnDG4zqnFoDK0oqzEngwNyjwQzjJEzDK2zDCNajGOnDeO)
is an orthonormal basis for euclidean n-space, and
is considered to be embedded in a Clifford algebra.
This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a spinor bundle.
- For a spin manifold, M, the Atiyah-Singer-Dirac operator is locally defined as follows: For
and
a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is
,
where
is a lifting of the Levi-Civita connection on M to the spinor bundle over M.
See also