Basis for a tensor product of Fock spaces
$begingroup$
Let $H, K$ be two Hilbert spaces with orthonormal bases ${e_{alpha}}, {f_{beta}}$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $H$, denoted as $mathscr{F}_-(H)$ (i.e. the anti-symmetrised Fock space), is given by
$$a^{dagger}_H(e_{alpha_1}) cdots a^{dagger}_H(e_{alpha_1}) Omega_H, $$
when ${e_{alpha_1}, ldots, e_{alpha_n}}$ runs over finite subsets of ${e_{alpha}}$. In the above, $a^{dagger}_H$ and $Omega_H$ are the creation operators and vacuum vector on $mathscr{F}_-(H)$, respectively. Now, I want to build an orthonormal basis for the space $mathscr{F}_-(H) otimes mathscr{F}_-(K)$. My first step to this was to formulate creation/annihilation operators on $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ that obey the CAR relations. These are given by
$$b^{dagger}(h,k) := frac{1}{sqrt{2}}left(a^{dagger}_H(h) otimes 1_K + left(1_Hright)^Notimes a_K^{dagger}(k)right),$$
where $hin H, kin K$ and $N$ is the particle number operator on $H$ and $1_H, 1_K$ are the identities on the subscripted spaces. I can show that all vectors of the form
$$b^{dagger}(e_{alpha_1}, f_{beta_1}) cdots b^{dagger}(e_{alpha_n}, f_{beta_n})Omega_Hotimes Omega_K$$
are orthonormal.$$$$
What I want to prove now is that the span of these vectors is dense in $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ when ${ (e_{alpha_1}, f_{beta_1}, ldots, (e_{alpha_n}, f_{beta_n})}$ runs over finite subsets of ${(e_{alpha}, f_{beta})}$. To me it makes sense that this should be the case since the set is built from the creation operators on the space, but I'm struggling to rigorously prove it.
functional-analysis hilbert-spaces mathematical-physics quantum-field-theory
edited 6 hours ago
Andrews
1,0141318
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
$endgroup$
add a comment |
$begingroup$
Let $H, K$ be two Hilbert spaces with orthonormal bases ${e_{alpha}}, {f_{beta}}$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $H$, denoted as $mathscr{F}_-(H)$ (i.e. the anti-symmetrised Fock space), is given by
$$a^{dagger}_H(e_{alpha_1}) cdots a^{dagger}_H(e_{alpha_1}) Omega_H, $$
when ${e_{alpha_1}, ldots, e_{alpha_n}}$ runs over finite subsets of ${e_{alpha}}$. In the above, $a^{dagger}_H$ and $Omega_H$ are the creation operators and vacuum vector on $mathscr{F}_-(H)$, respectively. Now, I want to build an orthonormal basis for the space $mathscr{F}_-(H) otimes mathscr{F}_-(K)$. My first step to this was to formulate creation/annihilation operators on $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ that obey the CAR relations. These are given by
$$b^{dagger}(h,k) := frac{1}{sqrt{2}}left(a^{dagger}_H(h) otimes 1_K + left(1_Hright)^Notimes a_K^{dagger}(k)right),$$
where $hin H, kin K$ and $N$ is the particle number operator on $H$ and $1_H, 1_K$ are the identities on the subscripted spaces. I can show that all vectors of the form
$$b^{dagger}(e_{alpha_1}, f_{beta_1}) cdots b^{dagger}(e_{alpha_n}, f_{beta_n})Omega_Hotimes Omega_K$$
are orthonormal.$$$$
What I want to prove now is that the span of these vectors is dense in $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ when ${ (e_{alpha_1}, f_{beta_1}, ldots, (e_{alpha_n}, f_{beta_n})}$ runs over finite subsets of ${(e_{alpha}, f_{beta})}$. To me it makes sense that this should be the case since the set is built from the creation operators on the space, but I'm struggling to rigorously prove it.
functional-analysis hilbert-spaces mathematical-physics quantum-field-theory
edited 6 hours ago
Andrews
1,0141318
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
$endgroup$
$begingroup$
Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way.
$endgroup$
– lcv
Dec 14 '18 at 23:39
$begingroup$
I'm looking for the former case, but with particular connection to the creation operators of the space
$endgroup$
– CS1994
Dec 15 '18 at 12:26
add a comment |
$begingroup$
Let $H, K$ be two Hilbert spaces with orthonormal bases ${e_{alpha}}, {f_{beta}}$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $H$, denoted as $mathscr{F}_-(H)$ (i.e. the anti-symmetrised Fock space), is given by
$$a^{dagger}_H(e_{alpha_1}) cdots a^{dagger}_H(e_{alpha_1}) Omega_H, $$
when ${e_{alpha_1}, ldots, e_{alpha_n}}$ runs over finite subsets of ${e_{alpha}}$. In the above, $a^{dagger}_H$ and $Omega_H$ are the creation operators and vacuum vector on $mathscr{F}_-(H)$, respectively. Now, I want to build an orthonormal basis for the space $mathscr{F}_-(H) otimes mathscr{F}_-(K)$. My first step to this was to formulate creation/annihilation operators on $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ that obey the CAR relations. These are given by
$$b^{dagger}(h,k) := frac{1}{sqrt{2}}left(a^{dagger}_H(h) otimes 1_K + left(1_Hright)^Notimes a_K^{dagger}(k)right),$$
where $hin H, kin K$ and $N$ is the particle number operator on $H$ and $1_H, 1_K$ are the identities on the subscripted spaces. I can show that all vectors of the form
$$b^{dagger}(e_{alpha_1}, f_{beta_1}) cdots b^{dagger}(e_{alpha_n}, f_{beta_n})Omega_Hotimes Omega_K$$
are orthonormal.$$$$
What I want to prove now is that the span of these vectors is dense in $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ when ${ (e_{alpha_1}, f_{beta_1}, ldots, (e_{alpha_n}, f_{beta_n})}$ runs over finite subsets of ${(e_{alpha}, f_{beta})}$. To me it makes sense that this should be the case since the set is built from the creation operators on the space, but I'm struggling to rigorously prove it.
functional-analysis hilbert-spaces mathematical-physics quantum-field-theory
edited 6 hours ago
Andrews
1,0141318
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
$endgroup$
Let $H, K$ be two Hilbert spaces with orthonormal bases ${e_{alpha}}, {f_{beta}}$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $H$, denoted as $mathscr{F}_-(H)$ (i.e. the anti-symmetrised Fock space), is given by
$$a^{dagger}_H(e_{alpha_1}) cdots a^{dagger}_H(e_{alpha_1}) Omega_H, $$
when ${e_{alpha_1}, ldots, e_{alpha_n}}$ runs over finite subsets of ${e_{alpha}}$. In the above, $a^{dagger}_H$ and $Omega_H$ are the creation operators and vacuum vector on $mathscr{F}_-(H)$, respectively. Now, I want to build an orthonormal basis for the space $mathscr{F}_-(H) otimes mathscr{F}_-(K)$. My first step to this was to formulate creation/annihilation operators on $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ that obey the CAR relations. These are given by
$$b^{dagger}(h,k) := frac{1}{sqrt{2}}left(a^{dagger}_H(h) otimes 1_K + left(1_Hright)^Notimes a_K^{dagger}(k)right),$$
where $hin H, kin K$ and $N$ is the particle number operator on $H$ and $1_H, 1_K$ are the identities on the subscripted spaces. I can show that all vectors of the form
$$b^{dagger}(e_{alpha_1}, f_{beta_1}) cdots b^{dagger}(e_{alpha_n}, f_{beta_n})Omega_Hotimes Omega_K$$
are orthonormal.$$$$
What I want to prove now is that the span of these vectors is dense in $mathscr{F}_-(H) otimes mathscr{F}_-(K)$ when ${ (e_{alpha_1}, f_{beta_1}, ldots, (e_{alpha_n}, f_{beta_n})}$ runs over finite subsets of ${(e_{alpha}, f_{beta})}$. To me it makes sense that this should be the case since the set is built from the creation operators on the space, but I'm struggling to rigorously prove it.
functional-analysis hilbert-spaces mathematical-physics quantum-field-theory
functional-analysis hilbert-spaces mathematical-physics quantum-field-theory
edited 6 hours ago
Andrews
1,0141318
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
edited 6 hours ago
Andrews
1,0141318
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
edited 6 hours ago
Andrews
1,0141318
edited 6 hours ago
Andrews
1,0141318
edited 6 hours ago
Andrews
1,0141318
1,0141318
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
asked Dec 14 '18 at 16:04
CS1994CS1994
1515
1515
$begingroup$
Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way.
$endgroup$
– lcv
Dec 14 '18 at 23:39
$begingroup$
I'm looking for the former case, but with particular connection to the creation operators of the space
$endgroup$
– CS1994
Dec 15 '18 at 12:26
add a comment |
$begingroup$
Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way.
$endgroup$
– lcv
Dec 14 '18 at 23:39
$begingroup$
I'm looking for the former case, but with particular connection to the creation operators of the space
$endgroup$
– CS1994
Dec 15 '18 at 12:26
$begingroup$
Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way.
$endgroup$
– lcv
Dec 14 '18 at 23:39
$begingroup$
Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way.
$endgroup$
– lcv
Dec 14 '18 at 23:39
$begingroup$
I'm looking for the former case, but with particular connection to the creation operators of the space
$endgroup$
– CS1994
Dec 15 '18 at 12:26
$begingroup$
I'm looking for the former case, but with particular connection to the creation operators of the space
$endgroup$
– CS1994
Dec 15 '18 at 12:26
add a comment |
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$begingroup$
Do you want a basis (a dense set of linearly independent vectors) for the tensor product of the Fock spaces or for the Fock space of the tensor product?. In the former case orthonormal sets are constructed in the usual way.
$endgroup$
– lcv
Dec 14 '18 at 23:39
$begingroup$
I'm looking for the former case, but with particular connection to the creation operators of the space
$endgroup$
– CS1994
Dec 15 '18 at 12:26