Antisymmetric subspace - Revision history

Nathaniel at 03:27, 22 November 2012

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Nathaniel: Created page with "The '''antisymmetric subspace''' $\mathcal{A}_p^d$ is the subspace of $(\mathbb{C}^d)^{\otimes p}$ of all vectors that are negated by odd permutations: : $\displaystyle \math..."

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Created page with "The '''antisymmetric subspace''' $\mathcal{A}_p^d$ is the subspace of $(\mathbb{C}^d)^{\otimes p}$ of all vectors that are negated by odd permutations: : $\displaystyle \math..."

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The '''antisymmetric subspace''' $\mathcal{A}_p^d$ is the subspace of $(\mathbb{C}^d)^{\otimes p}$ of all vectors that are negated by odd permutations:

: $\displaystyle \mathcal{A}_p^d \triangleq \big\{ \mathbf{v} \in (\mathbb{C}^d)^{\otimes p} : \mathbf{v} = (-1)^{{\rm sgn}(\sigma)}P_\sigma \mathbf{v} \ \ \forall \sigma \in S_p \big\},$

where $S_p$ is the [http://en.wikipedia.org/wiki/Symmetric_group symmetric group], ${\rm sgn}(\sigma)$ is the [http://en.wikipedia.org/wiki/Parity_of_a_permutation parity] of the permutation $\sigma$, and $P_\sigma$ is the unitary operator that permutes the $p$ subsystems according to the permutation $\sigma$:
: $\displaystyle P_\sigma(\mathbf{v}_1 \otimes \cdots \otimes \mathbf{v}_p) \triangleq \mathbf{v}_{\sigma(1)} \otimes \cdots \otimes \mathbf{v}_{\sigma(p)}$.

The antisymmetric subspace plays a role quite complementary to that of the [[symmetric subspace]] (and indeed, if $\mathcal{S}_p^d$ is the symmetric subspace then $\mathcal{A}_p^d \perp \mathcal{S}_p^d$).

==Construction==
The projection $P_{\mathcal{A}}$ onto the antisymmetric subspace can be constructed by averaging the signed permutation operators:<ref>R.B. Griffiths. [http://quantum.phys.cmu.edu/qm2/qmc161.pdf Systems of Identical Particles], ''33-756 Quantum Mechanics II Course Notes'', 2011.</ref>
: $\displaystyle P_{\mathcal{A}} = \frac{1}{p!}\sum_{\sigma \in S_p} (-1)^{{\rm sgn}(\sigma)}P_\sigma$.

In order to explicitly construct an orthonormal basis of the antisymmetric subspace, first fix an orthonormal basis $\{|i\rangle\}_{i=1}^d$ of $\mathbb{C}^d$. Now let $q$ be an increasing vector with $p$ entries containing distinct elements of $\{1, 2, \ldots, d\}$ as its entries. Then define
: $\displaystyle \mathbf{v}_q \triangleq \frac{1}{\sqrt{p!}}\sum_{\sigma \in S_p} (-1)^{{\rm sgn}(\sigma)} P_\sigma (| q_1 \rangle \otimes \cdots \otimes | q_p \rangle)$.
It is then the case that if $q$ ranges over all $\binom{d}{p}$ possible vectors $q$ then $\{\mathbf{v}_q\}_q$ is an orthonormal basis of $\mathcal{A}_p^d$.

==Facts==
* The dimension of the antisymmetric subspace is $\binom{d}{p}$. It follows that the antisymmetric subspace only has nonzero dimension when d ≥ p. When p = 2, the dimension of the antisymmetric subspace is d(d-1)/2, from which it follows that the symmetric and antisymmetric subspaces span the whole space in this case (and this case only): $\mathcal{A}_2^d \oplus \mathcal{S}_2^d \cong \mathbb{C}^d \otimes \mathbb{C}^d$.

==Related [[QETLAB]] functions==
* <tt>[[AntisymmetricProjection]]</tt>: Produces the [[projection]] onto the antisymmetric subspace $P_{\mathcal{A}}$
* <tt>[[PermutationOperator]]</tt>: Produces the permutation operator $P_\sigma$

==References==
<references />

@@ Line 9: / Line 9: @@
 ==Construction==
-The projection $P_{\mathcal{A}}$ onto the antisymmetric subspace can be constructed by averaging the signed permutation operators:<ref>R.B. Griffiths. [http://quantum.phys.cmu.edu/qm2/qmc161.pdf Systems of Identical Particles], ''33-756 Quantum Mechanics II Course Notes'', 2011.</ref>
+The [[orthogonal projection]] $P_{\mathcal{A}}$ onto the antisymmetric subspace can be constructed by averaging the signed permutation operators:<ref>R.B. Griffiths. [http://quantum.phys.cmu.edu/qm2/qmc161.pdf Systems of Identical Particles], ''33-756 Quantum Mechanics II Course Notes'', 2011.</ref>
 : $\displaystyle P_{\mathcal{A}} = \frac{1}{p!}\sum_{\sigma \in S_p} (-1)^{{\rm sgn}(\sigma)}P_\sigma$.
 In order to explicitly construct an orthonormal basis of the antisymmetric subspace, first fix an orthonormal basis $\{|i\rangle\}_{i=1}^d$ of $\mathbb{C}^d$. Now let $q$ be an increasing vector with $p$ entries containing distinct elements of $\{1, 2, \ldots, d\}$ as its entries. Then define

@@ Line 14: / Line 14: @@
 In order to explicitly construct an orthonormal basis of the antisymmetric subspace, first fix an orthonormal basis $\{|i\rangle\}_{i=1}^d$ of $\mathbb{C}^d$. Now let $q$ be an increasing vector with $p$ entries containing distinct elements of $\{1, 2, \ldots, d\}$ as its entries. Then define
 : $\displaystyle \mathbf{v}_q \triangleq \frac{1}{\sqrt{p!}}\sum_{\sigma \in S_p} (-1)^{{\rm sgn}(\sigma)} P_\sigma (| q_1 \rangle \otimes \cdots \otimes | q_p \rangle)$.
-It is then the case that if $q$ ranges over all $\binom{d}{p}$ possible vectors $q$ then $\{\mathbf{v}_q\}_q$ is an orthonormal basis of $\mathcal{A}_p^d$.
+It is then the case that $\{\mathbf{v}_q\}_q$ is an orthonormal basis of $\mathcal{A}_p^d$ if $q$ ranges over all $\binom{d}{p}$ possible increasing vectors with $p$ distinct entries chosen from $\{1, 2, \ldots, d\}$.
 ==Facts==

@@ Line 21: / Line 21: @@
 ==Related [[QETLAB]] functions==
 * <tt>[[AntisymmetricProjection]]</tt>: Produces the [[projection]] onto the antisymmetric subspace $P_{\mathcal{A}}$
-* <tt>[[PermutationOperator]]</tt>: Produces the permutation operator $P_\sigma$
+* <tt>[[PermutationOperator]]</tt>: Produces a permutation operator $P_\sigma$
 ==See also==