Mike Carroll
Jul21-04, 05:02 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\nBirkhoff & von Neumann\'s quantum logic [BvN] looks rather odd to a\nlogician.\n\nIn 2-valued logic, we have various valid formulas. These include for\nexample the law of the excluded middle, distributive laws for\nconjunction and disjunction, and so on. From the logical point of\nview, these are all equal: logic provides no criterion for determining\nthat some are more important than others.\n\nBirkhoff & von Neumann\'s system, however, does make a distinction\namong these valid formulas. Their quantum logic preserves the excluded\nmiddle, for example, but not the distributive laws for conjunction and\ndisjunction. This is why their system looks odd to a logician.\n\nIn fact, the logician is inclined to conclude from this oddity that\nBirkhoff & von Neumann made a mistake somewhere. Poking around in\ntheir mapping between logical operations and operations on linear\nsubspaces of a Hilbert space, he finds especially suspicious their\ndefinition of disjunction. The natural mapping would be to take the\ndisjunction of the propositions as the linear sum of the vectors in\nthe two subspaces. But this won\'t work, because the linear sum of 2\nclosed linear subspaces need not be closed, so the disjunction would\nnot be a quantum proposition. They dodge this by declaring that the\ndisjunction is the closure of the sum (cf. [Isham] p.207). The\nlogician suspects that this is one reason they end up with\nnon-standard behavior for disjunction.\n\nIs there a remedy?\n\nSuppose one maps a quantum proposition to a linear subspace, without\nrequiring that it be closed. Then all projection operators, being\nclosed linear subspaces, are propositions, but the converse does not\nhold. One has as one\'s quantum propositions a superset of the entities\nBirkhoff & von Neumann selected. One may then map disjunction to\nlinear sum.\n\nOf course, one then has a new set of problems. Some quantum\npropositions have 0 and 1 as eigenvalues, while others do not, so one\nmay have trouble if one wants to preserve 2-valued logic. But maybe\nthis collection looks more like what one would expect quantum\npropositions to look like anyway?\n\n[Reichenbach] investigated modifying 2-valued logic for quantum\nmechanics, but has no reference to [BvN].\n\nAFAIK no one has explored this path, but if anyone knows of any\nreferences in the literature I\'d appreciate hearing about them. I\nalso need to learn more about linear subspaces that are not closed,\nand would appreciate hearing about a standard math text that\'s strong\nin this area.\n\nMike Carroll\nOro Valley, AZ\n\n=====\n[BvN] G. Birkhoff and J. von Neumann, "The Logic of Quantum\nMechanics", Annals of Mathematics v.37 (1936), 823-43.\n[Isham] Chris Isham, "Lectures on Quantum Theory", Imperial College\nPress, 1995.\n[Reichenbach] Hans Reichenbach, "Philosophic Foundations of Quantum\nMechanics", 1944, Dover reprint 1998.\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>Birkhoff & von Neumann's quantum logic [BvN] looks rather odd to a
logician.
In 2-valued logic, we have various valid formulas. These include for
example the law of the excluded middle, distributive laws for
conjunction and disjunction, and so on. From the logical point of
view, these are all equal: logic provides no criterion for determining
that some are more important than others.
Birkhoff & von Neumann's system, however, does make a distinction
among these valid formulas. Their quantum logic preserves the excluded
middle, for example, but not the distributive laws for conjunction and
disjunction. This is why their system looks odd to a logician.
In fact, the logician is inclined to conclude from this oddity that
Birkhoff & von Neumann made a mistake somewhere. Poking around in
their mapping between logical operations and operations on linear
subspaces of a Hilbert space, he finds especially suspicious their
definition of disjunction. The natural mapping would be to take the
disjunction of the propositions as the linear sum of the vectors in
the two subspaces. But this won't work, because the linear sum of 2
closed linear subspaces need not be closed, so the disjunction would
not be a quantum proposition. They dodge this by declaring that the
disjunction is the closure of the sum (cf. [Isham] p.207). The
logician suspects that this is one reason they end up with
non-standard behavior for disjunction.
Is there a remedy?
Suppose one maps a quantum proposition to a linear subspace, without
requiring that it be closed. Then all projection operators, being
closed linear subspaces, are propositions, but the converse does not
hold. One has as one's quantum propositions a superset of the entities
Birkhoff & von Neumann selected. One may then map disjunction to
linear sum.
Of course, one then has a new set of problems. Some quantum
propositions have and 1 as eigenvalues, while others do not, so one
may have trouble if one wants to preserve 2-valued logic. But maybe
this collection looks more like what one would expect quantum
propositions to look like anyway?
[Reichenbach] investigated modifying 2-valued logic for quantum
mechanics, but has no reference to [BvN].
AFAIK no one has explored this path, but if anyone knows of any
references in the literature I'd appreciate hearing about them. I
also need to learn more about linear subspaces that are not closed,
and would appreciate hearing about a standard math text that's strong
in this area.
Mike Carroll
Oro Valley, AZ
=====
[BvN] G. Birkhoff and J. von Neumann, "The Logic of Quantum
Mechanics", Annals of Mathematics v.37 (1936), 823-43.
[Isham] Chris Isham, "Lectures on Quantum Theory", Imperial College
Press, 1995.
[Reichenbach] Hans Reichenbach, "Philosophic Foundations of Quantum
Mechanics", 1944, Dover reprint 1998.
logician.
In 2-valued logic, we have various valid formulas. These include for
example the law of the excluded middle, distributive laws for
conjunction and disjunction, and so on. From the logical point of
view, these are all equal: logic provides no criterion for determining
that some are more important than others.
Birkhoff & von Neumann's system, however, does make a distinction
among these valid formulas. Their quantum logic preserves the excluded
middle, for example, but not the distributive laws for conjunction and
disjunction. This is why their system looks odd to a logician.
In fact, the logician is inclined to conclude from this oddity that
Birkhoff & von Neumann made a mistake somewhere. Poking around in
their mapping between logical operations and operations on linear
subspaces of a Hilbert space, he finds especially suspicious their
definition of disjunction. The natural mapping would be to take the
disjunction of the propositions as the linear sum of the vectors in
the two subspaces. But this won't work, because the linear sum of 2
closed linear subspaces need not be closed, so the disjunction would
not be a quantum proposition. They dodge this by declaring that the
disjunction is the closure of the sum (cf. [Isham] p.207). The
logician suspects that this is one reason they end up with
non-standard behavior for disjunction.
Is there a remedy?
Suppose one maps a quantum proposition to a linear subspace, without
requiring that it be closed. Then all projection operators, being
closed linear subspaces, are propositions, but the converse does not
hold. One has as one's quantum propositions a superset of the entities
Birkhoff & von Neumann selected. One may then map disjunction to
linear sum.
Of course, one then has a new set of problems. Some quantum
propositions have and 1 as eigenvalues, while others do not, so one
may have trouble if one wants to preserve 2-valued logic. But maybe
this collection looks more like what one would expect quantum
propositions to look like anyway?
[Reichenbach] investigated modifying 2-valued logic for quantum
mechanics, but has no reference to [BvN].
AFAIK no one has explored this path, but if anyone knows of any
references in the literature I'd appreciate hearing about them. I
also need to learn more about linear subspaces that are not closed,
and would appreciate hearing about a standard math text that's strong
in this area.
Mike Carroll
Oro Valley, AZ
=====
[BvN] G. Birkhoff and J. von Neumann, "The Logic of Quantum
Mechanics", Annals of Mathematics v.37 (1936), 823-43.
[Isham] Chris Isham, "Lectures on Quantum Theory", Imperial College
Press, 1995.
[Reichenbach] Hans Reichenbach, "Philosophic Foundations of Quantum
Mechanics", 1944, Dover reprint 1998.