Thomas Larsson
Oct20-04, 03:18 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nAs a response to Urs in another forum, let me describe how phase space\ncan be described in a covariant way.\n\nA phase space has local coordinates (q, p). Time evolution is given\nby Hamilton\'s equations, dq/dt = {q, H}, dp/dt = {p, H}, where H is\nthe Hamiltonian. The solution to these equations is some curve\n(q(t), p(t)). Since the curve is completely specified by the pair\n(q, p) = (q(0), p(0)), there is a 1-1 correspondence between solutions\nto the equations of motion and points in phase space.\n\nWe can therefore identify phase space with the space of curves solving\nHamilton\'s equation. This is the covariant phase space. The Hamiltonian\ntakes a very simple form in this formalism; it is simply the generator of\nt-translations. However, we have of course not gained anything from a\npractical point of view, because describing these curves is equivalent to\nsolving Hamilton\'s equations.\n\nHowever, we are not mainly interested in phase space itself, but rather in\nthe space of functions over it, C(q,p). It is this function space which\ncorresponds to the Hilbert space after quantization. C(q,p) can be easily\ndescribed as the space of arbitrary functionals C[q(t),p(t)], modulo the\nideal generated by the equations of motion.\n\nActually, people usually ignore p(t) because it is fully specified by the\nEuler-Lagrange equations, symbolically written as E(t) = 0. These equation\ngenerate an ideal I in the space of functionals C[q(t)]. The ideal I\nconsistent of functionals of the form E(t) f[q(t)], f arbitrary, and we want\nto construct the space C[q(t)]/I. This space can be obtained as the zeroth\ncohomology group of a certain complex, called the Koszul-Tate complex in\nmath. Cohomology and complexes are scary, I know, but people who know about\nthe BRST complex will hopefully still follow.\n\nSo we introduce a fermionic antifield q*(t) and a KT differential d defined\nby\n\nd q(t) = 0, d q*(t) = E(t).\n\nThe enlarged function space C[q(t),q*(t)] can be decomposed into\nsubspace V_i of fixed antifield number. Since the KT differential obviously\nis nilpotent (E(t) depends on q(t) but not on q*(t)), it defines a complex\n\n0 <-- V_0 <-- V_1 <-- V_2 <-- ... .\n\nThe cohomology space are defined as usual as H_i = ker d/im d. It is a\nstandard proof that this complex yields a resolution of C[q(t)]/I, i.e.\nthat H_0 = C[q(t)]/I and H_i = 0 for all i > 0.\n\nLet us now return to the full phase space by adding the momenta p(t) and\np*(t), satisfying\n\n[p(t), q(t\')] = delta(t-t\'), [p*(t), q*(t\')]_+ = delta(t-t\').\n\nThe KT differential acts on arbitary functional F(q,q*,p,p*) as dF = [Q, F],\nwhere\n\nQ = \\int dt E(t) p*(t).\n\nWe can again introduce an enlarged function space C[q(t),q*(t),p(t),p*(t)]\nwhich decomposes into subspaces of fixed antifield number. The complex now\nextends to infinity in both directions,\n\n... V_-2 <-- V_-1 <-- V_0 <-- V_1 <-- V_2 <-- ... .\n\nAgain we identify the zeroth cohomology group H^0(Q) as the space of\nfunctions over the physical phase space.\n\nNow let\'s quantize! The idea is that if we quantize each space V_i, then the\ncohomology group H^0(Q) is the Hilbert space (or pre-Hilbert spaces, since\nwe don\'t talk about an inner product) of the quantum theory with Euler-\nLagrange equation E(t) = 0.\n\nSo, what about diffeomorphisms? Assume that the dynamics is diff invariant,\nas in GR. Then the unrestricted spaces V_i are spaces of tensor fields,\nand we know how to quantize them, making all diffeomorphisms into\nwell-defined operators. This is the stuff about passing to the Taylor data\nand normal ordering. Moreover, the KT operator Q is diff invariant, even\nafter quantization. It is important to notice here that Q is already\nnormal ordered, because E(t) and p*(t) commute. Therefore, it always\nremains nilpotent after quantization, and the cohomology groups will\ncarry representations of the anomalous diffeomorphism algebra.\n\nWe now see way the physical 4-diff symmetry becomes the Dirac algebra in\nconventional canonical quantization. We label phase space by points (q,p) =\n(q(0),p(0)). The result of some 4-diffs, e.g. time translations, lies\noutside points of this form. We must therefore add a compensating\ntransformation to get back to a point of the form (q\'(0),p\'(0)). But this\nis only an artefact of our choice of coordinatization and nothing physical.\nPhysically, the Dirac algebra and the 4-diff algebra are equivalent.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>As a response to Urs in another forum, let me describe how phase space
can be described in a covariant way.
A phase space has local coordinates (q, p). Time evolution is given
by Hamilton's equations, dq/dt = {q, H}, dp/dt = {p, H}, where H is
the Hamiltonian. The solution to these equations is some curve
(q(t), p(t)). Since the curve is completely specified by the pair
(q, p) = (q(0), p(0)), there is a 1-1 correspondence between solutions
to the equations of motion and points in phase space.
We can therefore identify phase space with the space of curves solving
Hamilton's equation. This is the covariant phase space. The Hamiltonian
takes a very simple form in this formalism; it is simply the generator of
t-translations. However, we have of course not gained anything from a
practical point of view, because describing these curves is equivalent to
solving Hamilton's equations.
However, we are not mainly interested in phase space itself, but rather in
the space of functions over it, C(q,p). It is this function space which
corresponds to the Hilbert space after quantization. C(q,p) can be easily
described as the space of arbitrary functionals C[q(t),p(t)], modulo the
ideal generated by the equations of motion.
Actually, people usually ignore p(t) because it is fully specified by the
Euler-Lagrange equations, symbolically written as E(t) = . These equation
generate an ideal I in the space of functionals C[q(t)]. The ideal I
consistent of functionals of the form E(t) f[q(t)], f arbitrary, and we want
to construct the space C[q(t)]/I. This space can be obtained as the zeroth
cohomology group of a certain complex, called the Koszul-Tate complex in
math. Cohomology and complexes are scary, I know, but people who know about
the BRST complex will hopefully still follow.
So we introduce a fermionic antifield q*(t) and a KT differential d defined
by
d q(t) = 0, d q*(t) = E(t).
The enlarged function space C[q(t),q*(t)] can be decomposed into
subspace V_i of fixed antifield number. Since the KT differential obviously
is nilpotent (E(t) depends on q(t) but not on q*(t)), it defines a complex
<-- V_0 <-- V_1 <-- V_2 <-- ... .
The cohomology space are defined as usual as H_i = ker d/im d. It is a
standard proof that this complex yields a resolution of C[q(t)]/I, i.e.
that H_0 = C[q(t)]/I and H_i = for all i > .
Let us now return to the full phase space by adding the momenta p(t) and
p*(t), satisfying
[p(t), q(t')] = \delta(t-t'), [p*(t), q*(t')]_+ = \delta(t-t').
The KT differential acts on arbitary functional F(q,q*,p,p*) as dF = [Q, F],
where
Q = \int dt E(t) p*(t).
We can again introduce an enlarged function space C[q(t),q*(t),p(t),p*(t)]
which decomposes into subspaces of fixed antifield number. The complex now
extends to infinity in both directions,
... V_-2 <-- V_-1 <-- V_0 <-- V_1 <-- V_2 <-- ... .
Again we identify the zeroth cohomology group H^0(Q) as the space of
functions over the physical phase space.
Now let's quantize! The idea is that if we quantize each space V_i, then the
cohomology group H^0(Q) is the Hilbert space (or pre-Hilbert spaces, since
we don't talk about an inner product) of the quantum theory with Euler-
Lagrange equation E(t) = .
So, what about diffeomorphisms? Assume that the dynamics is diff invariant,
as in GR. Then the unrestricted spaces V_i are spaces of tensor fields,
and we know how to quantize them, making all diffeomorphisms into
well-defined operators. This is the stuff about passing to the Taylor data
and normal ordering. Moreover, the KT operator Q is diff invariant, even
after quantization. It is important to notice here that Q is already
normal ordered, because E(t) and p*(t) commute. Therefore, it always
remains nilpotent after quantization, and the cohomology groups will
carry representations of the anomalous diffeomorphism algebra.
We now see way the physical 4-diff symmetry becomes the Dirac algebra in
conventional canonical quantization. We label phase space by points (q,p) =
(q(0),p(0)). The result of some 4-diffs, e.g. time translations, lies
outside points of this form. We must therefore add a compensating
transformation to get back to a point of the form (q'(0),p'(0)). But this
is only an artefact of our choice of coordinatization and nothing physical.
Physically, the Dirac algebra and the 4-diff algebra are equivalent.
can be described in a covariant way.
A phase space has local coordinates (q, p). Time evolution is given
by Hamilton's equations, dq/dt = {q, H}, dp/dt = {p, H}, where H is
the Hamiltonian. The solution to these equations is some curve
(q(t), p(t)). Since the curve is completely specified by the pair
(q, p) = (q(0), p(0)), there is a 1-1 correspondence between solutions
to the equations of motion and points in phase space.
We can therefore identify phase space with the space of curves solving
Hamilton's equation. This is the covariant phase space. The Hamiltonian
takes a very simple form in this formalism; it is simply the generator of
t-translations. However, we have of course not gained anything from a
practical point of view, because describing these curves is equivalent to
solving Hamilton's equations.
However, we are not mainly interested in phase space itself, but rather in
the space of functions over it, C(q,p). It is this function space which
corresponds to the Hilbert space after quantization. C(q,p) can be easily
described as the space of arbitrary functionals C[q(t),p(t)], modulo the
ideal generated by the equations of motion.
Actually, people usually ignore p(t) because it is fully specified by the
Euler-Lagrange equations, symbolically written as E(t) = . These equation
generate an ideal I in the space of functionals C[q(t)]. The ideal I
consistent of functionals of the form E(t) f[q(t)], f arbitrary, and we want
to construct the space C[q(t)]/I. This space can be obtained as the zeroth
cohomology group of a certain complex, called the Koszul-Tate complex in
math. Cohomology and complexes are scary, I know, but people who know about
the BRST complex will hopefully still follow.
So we introduce a fermionic antifield q*(t) and a KT differential d defined
by
d q(t) = 0, d q*(t) = E(t).
The enlarged function space C[q(t),q*(t)] can be decomposed into
subspace V_i of fixed antifield number. Since the KT differential obviously
is nilpotent (E(t) depends on q(t) but not on q*(t)), it defines a complex
<-- V_0 <-- V_1 <-- V_2 <-- ... .
The cohomology space are defined as usual as H_i = ker d/im d. It is a
standard proof that this complex yields a resolution of C[q(t)]/I, i.e.
that H_0 = C[q(t)]/I and H_i = for all i > .
Let us now return to the full phase space by adding the momenta p(t) and
p*(t), satisfying
[p(t), q(t')] = \delta(t-t'), [p*(t), q*(t')]_+ = \delta(t-t').
The KT differential acts on arbitary functional F(q,q*,p,p*) as dF = [Q, F],
where
Q = \int dt E(t) p*(t).
We can again introduce an enlarged function space C[q(t),q*(t),p(t),p*(t)]
which decomposes into subspaces of fixed antifield number. The complex now
extends to infinity in both directions,
... V_-2 <-- V_-1 <-- V_0 <-- V_1 <-- V_2 <-- ... .
Again we identify the zeroth cohomology group H^0(Q) as the space of
functions over the physical phase space.
Now let's quantize! The idea is that if we quantize each space V_i, then the
cohomology group H^0(Q) is the Hilbert space (or pre-Hilbert spaces, since
we don't talk about an inner product) of the quantum theory with Euler-
Lagrange equation E(t) = .
So, what about diffeomorphisms? Assume that the dynamics is diff invariant,
as in GR. Then the unrestricted spaces V_i are spaces of tensor fields,
and we know how to quantize them, making all diffeomorphisms into
well-defined operators. This is the stuff about passing to the Taylor data
and normal ordering. Moreover, the KT operator Q is diff invariant, even
after quantization. It is important to notice here that Q is already
normal ordered, because E(t) and p*(t) commute. Therefore, it always
remains nilpotent after quantization, and the cohomology groups will
carry representations of the anomalous diffeomorphism algebra.
We now see way the physical 4-diff symmetry becomes the Dirac algebra in
conventional canonical quantization. We label phase space by points (q,p) =
(q(0),p(0)). The result of some 4-diffs, e.g. time translations, lies
outside points of this form. We must therefore add a compensating
transformation to get back to a point of the form (q'(0),p'(0)). But this
is only an artefact of our choice of coordinatization and nothing physical.
Physically, the Dirac algebra and the 4-diff algebra are equivalent.