Lubos Motl
Sep21-04, 08:02 PM
<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>Quantization without quantization - prejudices and reality\n========================================= =================\n\nLet me just write some additional comments about this mode of thinking\nthat I claim underlies LQG. Let me also write at the beginning that I am\nconvinced that every sentence below labeled by "LQG" is deeply flawed and\nexplanations will be given at the end.\n\nLQG meta-axiom I: Whatever is true for a classical theory T must also be\ntrue for its quantization Q(T). Quantization is a bijective functor that\nrelates objects on both sides by a one-to-one correspondence.\n\nLQG theorem II: We should always look for quantization Q(T) for any\ninteresting theory T. Q(T) always exists.\n\nLQG corollary IIa: The most interesting generalization of quantum\nmechanics is the search for a Hilbert space and operators corresponding to\na given classical phase space. The direction of progress is to consider\nincreasingly complicated phase spaces.\n\nLQG corollary IIb: The most interesting parts of physics are the tools\n(e.g. choice of variables) how to perform the quantization; they are more\nimportant than the ability of the theories to predict and the results\nthemselves. Examples of these tools are Ashtekar\'s variables, spin\nnetworks, and spin foam.\n\nLQG corollary IIc: Unusual properties of a quantum theory - such as\nnon-separable Hilbert space; non-existence of a path-integral formulation;\nabsence of doable (e.g. perturbative) calculations (of the S-matrix) do\nnot constitute a problem as long as the theory can be called Q(T) - which\nis the most important thing.\n\nLQG implication III: A quantum theory is as interesting as its classical\ncounterpart. An interesting quantum theory can only be obtained by\nquantizing a classical theory.\n\nLQG corollary IIIa: A direct quantization of pure GR should be attempted,\ndespite failures, because GR is a nice and simple classical theory.\n\nLQG corollary IV: A classical theory T has nice and transparent relations\nbetween the observable. Its quantization Q(T) should therefore have nice\nrelations between the observables, too.\n\nLQG corollary IVa: Nonzero anomalies are impossible. This includes the\ncentral extensions of the algebra, such as the central charge of a CFT.\nOne should assume that they are zero, and only look for a proof\n(representation) of this assumption.\n\nLQG corollary IVb: Ultraviolet divergences are impossible, too.\n\nLQG corollary IVc: The non-renormalizable divergences can only appear if\nthe quantization Q(T) is done incorrectly.\n\nLQG corollary IVd (optional): The uncertainty principle must be incorrect\nbecause it destroys the nice relations between the observables.\n\nOK, let\'s stop with the LQG crap now.\n\nReality I. A quantum theory is the most radical transformation of the\nconcepts of physics that we have seen so far, and it implies a lot of new\nand unexpected phenomena. The classical intuition is often incorrect in\nthe quantum world, and no one-to-one correspondence exists. Quantum\nmechanics defines new rules which theory are predictive, interesting, and\nconsistent - rules whose majority is not contained in the classical limit.\n\nReality II. A classical theory often does not have a consistent quantum\ncounterpart. An attempt to quantize a theory is often spoiled by anomalies\nor non-renormalizable divergences. There are many interesting classical\ntheories (such as GR of Fermi\'s theory) that cannot be quantized. There\nare also many interesting quantum theories whose classical counterparts\nare not too interesting, for example because they are trivial\n(Chern-Simons theory). "Reality III" below offers more examples.\n\nReality IIa. General quantum mechanical systems with a Hamiltonian that is\nan arbitrary function on the phase space - which itself has an arbitrary\nshape and topology - have not been too important and interesting in\nobservable physics. Virtually every interesting application of quantum\nmechanics of point-like particles is based on wavefunctions on a\nwell-defined configuration space (x); the spectrum of the momentum\noperator is therefore easily calculable. The really interesting\nramifications of quantum mechanics involve different ideas than mixing of\nthe coordinates and the momenta and making the geometry of the space space\na bit more complicated. In useful applications, the coordinates and\nmomenta are kinematically decoupled, and one can view this fact as a\nnon-rigorous, simple version of a Coleman-Mandula-like theorem.\n\nThe phase spaces of complicated topology are not too revolutionary from\nanother reason: if they are small (small number of quantum states), the\nclassical-quantum link is too fuzzy and ambiguous. If they are large, on\nthe other hand, the "local" physics on the phase space reproduces physics\non infinite smooth phase space, and is therefore qualitatively understood.\n\nThe progress in physics in the last 70 years has been based on very\ndifferent ideas than convoluted phase spaces, especially the observed new\nways how relatively simple, "polynomial" systems of very well-defined\ndegrees of freedom can behave, and new local symmetries and their origin.\nThis includes gauge theories; Higgs mechanism; confinement; insights of\nthe renormalization group; generation of spacetime physics from the\nstringy worldsheet, and so on.\n\nReality IIb. In a consistent quantum theory with a classical limit, the\ndifferent procedures of quantization must be equivalent. A redefinition of\nvariables can help us to understand some phenomena, but it cannot make an\ninconsistent theory consistent. Finally, methods are only interesting if\nthey lead to interesting results, and a structure claimed to be Q(T) is\nnot necessarily an interesting result.\n\nReality IIc. A physically useful quantum theory must have a separable\nHilbert space - or at least the physically relevant sector must be\nseparable. A theory must be able to make some predictions that can be\nruled out, at least in principle. Whether or not a system of ideas can be\ncalled Q(T) for some classical theory T is not important physically. The\nnon-existence of the path-integral formulation or another usual approach\nto the theory indicates that the theory has some internal problems.\n\nReality III. The only success story in quantizing a known classical theory\nabove 1 dimension was QED. Electrodynamics was a classically studied\ntheory that happened to have a perturbatively consistent quantum\nextension. No other fundamental force is an example. Pure GR is\ninconsistent as a quantum theory. The weak sector of the electroweak\ntheory was found directly as a quantum theory; the virtual (quantum)\nW-bosons and the Z-bosons were proposed before the classical limit of the\nelectroweak theory (and physical, external gauge bosons, together with\ntheir vevs) were considered: another example in which the interesting\nquantum theory had to be found before its classical counterpart that is\nnot so interesting. Also, the equations of QCD were seriously proposed\nonly when it was known that the (quantum) beta function was negative and\nthe force was able to confine quarks and describe asymptotic freedom at\nhigh energies. The classical QCD was not an interesting theory either\nbecause it led to new, unobserved long-range forces, as Pauli proved\n(neglecting quantum mechanics in the field-theoretical context) in the\n1950s.\n\nThe question whether a quantum theory is interesting is largerly\nuncorrelated with the question whether its classical limit is interesting.\nExamples have been given in "Reality IIa". More generally, a quantum\ntheory is only interesting if it is predictive. It is very important for\nthe predictive power of a theory that a generic (high-order) interaction\nwould lead to inconsistencies in the UV. If it did not, infinitely many\ncoefficients would be undetermined. In consistent theories they can either\nbe neglected, or their coefficients\' smallness can be justified because\nthese are the irrelevant operators associated with new physics at much\nhigher scales, as the logic of the renormalization group dictates, and the\nsuper-high energy physics does not directly influence physics at\naccessible energies. A framework that would allow arbitrary couplings to\nbe added would imply a complete loss of predictive power. String theory is\nthe only known framework that constrains the parameters in a quantum\ntheory by mechanisms that go beyond the RG.\n\nThese are the physically relevant questions, the "landscape" of\ninteresting field theories is mapped by the Renormalization Group (RG),\nand these tools have nothing to do with the beauty of the classical limit.\n\nIncidentally, the beauty of a classical theory cannot be directly\nconnected with predictivity - because new, more complicated terms can\nessentially always be added - and therefore the "beauty" of a classical\ntheory is rather close to the subjective notion of simplicity whose direct\nphysical importance is very limited: a classical GR with R^2 terms is not\ninferior to the pure GR, except for purely subjective reasons. On the\nother hand, quantum physics of gravity gives a principle to label R^2 as\nthe less important term.\n\nOn the other hand, it is also believed that some interesting quantum\ntheories do not follow from a classical starting point. It is also known\nfor sure that a quantum theory can have several classical limits.\n\nReality IIIa: GR is an example of a non-renormalizable theory, and the\nproblematic UV behavior is a key to new physics that must regularize the\ndivergences. In the case of Fermi\'s theory, the UV completion turns out to\nbe essentially unique - a spontaneously broken gauge theory. General\ncovariance is as much constraining or more, and a UV completion must be\ntaken seriously. String theory is the only known solution how to generate\nconsistent loop (quantum) contributions to the scattering amplitudes.\n\nThe only special feature of GR is the general covariance, which is a\ngeneralization of gauge symmetries (with a spin greater by one). The\nstructure respecting this gauge symmetry is important for classical\ndecoupling of the unphysical modes - at the non-linear, multiparticle\nlevel - but it does not imply anything about the quantum loops of GR.\n\nReality IV: Quantum field theory implies that the observables do have\nshort distance divergences in their products. Algebras can acquire new\nterms that did not exist classically. This behavior can be measured in\nsome cases; in other cases it directly follows from a nonzero commutator\n(the uncertainty principle) and the equations of motion.\n\nReality IVa: Chiral gauge theories and similar theories in 2k dimensions\ntypically lead to gauge anomalies. Gauge anomalies imply that the theory\nis no longer consistent because the unphysical (and negative-norm)\npolarizations of the gauge bosons (or graviton) no longer decouple. The\nstructure of anomalies can be calculated if one tries to regularize\ncertain UV divergences, but the result is purely reflected by the low\nenergy (IR) spectrum of the theory. There is no way to avoid the\nconclusion about gauge anomalies except for cancelling them.\n\nAnomalies in global symmetries do not imply an inconsistency, but they\ndrastically influence the physics of a given theory.\n\nCentral charge (or Weyl anomaly) is a specific example of an anomalous\nterm that can be calculated in many ways, and there is no consistent way\nto avoid its nonzero value.\n\nReality IVb: The existence of UV divergences is real. This existence is in\nfact essential to our ability to set the coupling constants and other\nparameters equal to their observed values.\n\nReality IVc: Non-renormalizability of some UV divergences is not a\nspeculation, but a result of quantitative calculations. The only way how\nthese problems could disappear in a reformulated theory is the situation\nthat the given theory has a UV fixed point under the renormalization\ngroup. The non-trivial, interacting fixed points are very rare, and there\nexists no good reason why such a fixed point should exist and be\nassociated with a generic non-renormalizable theory.\n\nReality IVd (optional): The uncertainty principle is the basic principle\nbehind the whole structure of quantum physics and everything that contains\nthe adjective "quantum".\n______________________________________ ________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\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>Quantization without quantization - prejudices and reality
================================================== ========
Let me just write some additional comments about this mode of thinking
that I claim underlies LQG. Let me also write at the beginning that I am
convinced that every sentence below labeled by "LQG" is deeply flawed and
explanations will be given at the end.
LQG meta-axiom I: Whatever is true for a classical theory T must also be
true for its quantization Q(T). Quantization is a bijective functor that
relates objects on both sides by a one-to-one correspondence.
LQG theorem II: We should always look for quantization Q(T) for any
interesting theory T. Q(T) always exists.
LQG corollary IIa: The most interesting generalization of quantum
mechanics is the search for a Hilbert space and operators corresponding to
a given classical phase space. The direction of progress is to consider
increasingly complicated phase spaces.
LQG corollary IIb: The most interesting parts of physics are the tools
(e.g. choice of variables) how to perform the quantization; they are more
important than the ability of the theories to predict and the results
themselves. Examples of these tools are Ashtekar's variables, spin
networks, and spin foam.
LQG corollary IIc: Unusual properties of a quantum theory - such as
non-separable Hilbert space; non-existence of a path-integral formulation;
absence of doable (e.g. perturbative) calculations (of the S-matrix) do
not constitute a problem as long as the theory can be called Q(T) - which
is the most important thing.
LQG implication III: A quantum theory is as interesting as its classical
counterpart. An interesting quantum theory can only be obtained by
quantizing a classical theory.
LQG corollary IIIa: A direct quantization of pure GR should be attempted,
despite failures, because GR is a nice and simple classical theory.
LQG corollary IV: A classical theory T has nice and transparent relations
between the observable. Its quantization Q(T) should therefore have nice
relations between the observables, too.
LQG corollary IVa: Nonzero anomalies are impossible. This includes the
central extensions of the algebra, such as the central charge of a CFT.
One should assume that they are zero, and only look for a proof
(representation) of this assumption.
LQG corollary IVb: Ultraviolet divergences are impossible, too.
LQG corollary IVc: The non-renormalizable divergences can only appear if
the quantization Q(T) is done incorrectly.
LQG corollary IVd (optional): The uncertainty principle must be incorrect
because it destroys the nice relations between the observables.
OK, let's stop with the LQG crap now.
Reality I. A quantum theory is the most radical transformation of the
concepts of physics that we have seen so far, and it implies a lot of new
and unexpected phenomena. The classical intuition is often incorrect in
the quantum world, and no one-to-one correspondence exists. Quantum
mechanics defines new rules which theory are predictive, interesting, and
consistent - rules whose majority is not contained in the classical limit.
Reality II. A classical theory often does not have a consistent quantum
counterpart. An attempt to quantize a theory is often spoiled by anomalies
or non-renormalizable divergences. There are many interesting classical
theories (such as GR of Fermi's theory) that cannot be quantized. There
are also many interesting quantum theories whose classical counterparts
are not too interesting, for example because they are trivial
(Chern-Simons theory). "Reality III" below offers more examples.
Reality IIa. General quantum mechanical systems with a Hamiltonian that is
an arbitrary function on the phase space - which itself has an arbitrary
shape and topology - have not been too important and interesting in
observable physics. Virtually every interesting application of quantum
mechanics of point-like particles is based on wavefunctions on a
well-defined configuration space (x); the spectrum of the momentum
operator is therefore easily calculable. The really interesting
ramifications of quantum mechanics involve different ideas than mixing of
the coordinates and the momenta and making the geometry of the space space
a bit more complicated. In useful applications, the coordinates and
momenta are kinematically decoupled, and one can view this fact as a
non-rigorous, simple version of a Coleman-Mandula-like theorem.
The phase spaces of complicated topology are not too revolutionary from
another reason: if they are small (small number of quantum states), the
classical-quantum link is too fuzzy and ambiguous. If they are large, on
the other hand, the "local" physics on the phase space reproduces physics
on infinite smooth phase space, and is therefore qualitatively understood.
The progress in physics in the last 70 years has been based on very
different ideas than convoluted phase spaces, especially the observed new
ways how relatively simple, "polynomial" systems of very well-defined
degrees of freedom can behave, and new local symmetries and their origin.
This includes gauge theories; Higgs mechanism; confinement; insights of
the renormalization group; generation of spacetime physics from the
stringy worldsheet, and so on.
Reality IIb. In a consistent quantum theory with a classical limit, the
different procedures of quantization must be equivalent. A redefinition of
variables can help us to understand some phenomena, but it cannot make an
inconsistent theory consistent. Finally, methods are only interesting if
they lead to interesting results, and a structure claimed to be Q(T) is
not necessarily an interesting result.
Reality IIc. A physically useful quantum theory must have a separable
Hilbert space - or at least the physically relevant sector must be
separable. A theory must be able to make some predictions that can be
ruled out, at least in principle. Whether or not a system of ideas can be
called Q(T) for some classical theory T is not important physically. The
non-existence of the path-integral formulation or another usual approach
to the theory indicates that the theory has some internal problems.
Reality III. The only success story in quantizing a known classical theory
above 1 dimension was QED. Electrodynamics was a classically studied
theory that happened to have a perturbatively consistent quantum
extension. No other fundamental force is an example. Pure GR is
inconsistent as a quantum theory. The weak sector of the electroweak
theory was found directly as a quantum theory; the virtual (quantum)
W-bosons and the Z-bosons were proposed before the classical limit of the
electroweak theory (and physical, external gauge bosons, together with
their vevs) were considered: another example in which the interesting
quantum theory had to be found before its classical counterpart that is
not so interesting. Also, the equations of QCD were seriously proposed
only when it was known that the (quantum) \beta function was negative and
the force was able to confine quarks and describe asymptotic freedom at
high energies. The classical QCD was not an interesting theory either
because it led to new, unobserved long-range forces, as Pauli proved
(neglecting quantum mechanics in the field-theoretical context) in the
1950s.
The question whether a quantum theory is interesting is largerly
uncorrelated with the question whether its classical limit is interesting.
Examples have been given in "Reality IIa". More generally, a quantum
theory is only interesting if it is predictive. It is very important for
the predictive power of a theory that a generic (high-order) interaction
would lead to inconsistencies in the UV. If it did not, infinitely many
coefficients would be undetermined. In consistent theories they can either
be neglected, or their coefficients' smallness can be justified because
these are the irrelevant operators associated with new physics at much
higher scales, as the logic of the renormalization group dictates, and the
super-high energy physics does not directly influence physics at
accessible energies. A framework that would allow arbitrary couplings to
be added would imply a complete loss of predictive power. String theory is
the only known framework that constrains the parameters in a quantum
theory by mechanisms that go beyond the RG.
These are the physically relevant questions, the "landscape" of
interesting field theories is mapped by the Renormalization Group (RG),
and these tools have nothing to do with the beauty of the classical limit.
Incidentally, the beauty of a classical theory cannot be directly
connected with predictivity - because new, more complicated terms can
essentially always be added - and therefore the "beauty" of a classical
theory is rather close to the subjective notion of simplicity whose direct
physical importance is very limited: a classical GR with R^2 terms is not
inferior to the pure GR, except for purely subjective reasons. On the
other hand, quantum physics of gravity gives a principle to label R^2 as
the less important term.
On the other hand, it is also believed that some interesting quantum
theories do not follow from a classical starting point. It is also known
for sure that a quantum theory can have several classical limits.
Reality IIIa: GR is an example of a non-renormalizable theory, and the
problematic UV behavior is a key to new physics that must regularize the
divergences. In the case of Fermi's theory, the UV completion turns out to
be essentially unique - a spontaneously broken gauge theory. General
covariance is as much constraining or more, and a UV completion must be
taken seriously. String theory is the only known solution how to generate
consistent loop (quantum) contributions to the scattering amplitudes.
The only special feature of GR is the general covariance, which is a
generalization of gauge symmetries (with a spin greater by one). The
structure respecting this gauge symmetry is important for classical
decoupling of the unphysical modes - at the non-linear, multiparticle
level - but it does not imply anything about the quantum loops of GR.
Reality IV: Quantum field theory implies that the observables do have
short distance divergences in their products. Algebras can acquire new
terms that did not exist classically. This behavior can be measured in
some cases; in other cases it directly follows from a nonzero commutator
(the uncertainty principle) and the equations of motion.
Reality IVa: Chiral gauge theories and similar theories in 2k dimensions
typically lead to gauge anomalies. Gauge anomalies imply that the theory
is no longer consistent because the unphysical (and negative-norm)
polarizations of the gauge bosons (or graviton) no longer decouple. The
structure of anomalies can be calculated if one tries to regularize
certain UV divergences, but the result is purely reflected by the low
energy (IR) spectrum of the theory. There is no way to avoid the
conclusion about gauge anomalies except for cancelling them.
Anomalies in global symmetries do not imply an inconsistency, but they
drastically influence the physics of a given theory.
Central charge (or Weyl anomaly) is a specific example of an anomalous
term that can be calculated in many ways, and there is no consistent way
to avoid its nonzero value.
Reality IVb: The existence of UV divergences is real. This existence is in
fact essential to our ability to set the coupling constants and other
parameters equal to their observed values.
Reality IVc: Non-renormalizability of some UV divergences is not a
speculation, but a result of quantitative calculations. The only way how
these problems could disappear in a reformulated theory is the situation
that the given theory has a UV fixed point under the renormalization
group. The non-trivial, interacting fixed points are very rare, and there
exists no good reason why such a fixed point should exist and be
associated with a generic non-renormalizable theory.
Reality IVd (optional): The uncertainty principle is the basic principle
behind the whole structure of quantum physics and everything that contains
the adjective "quantum".
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
================================================== ========
Let me just write some additional comments about this mode of thinking
that I claim underlies LQG. Let me also write at the beginning that I am
convinced that every sentence below labeled by "LQG" is deeply flawed and
explanations will be given at the end.
LQG meta-axiom I: Whatever is true for a classical theory T must also be
true for its quantization Q(T). Quantization is a bijective functor that
relates objects on both sides by a one-to-one correspondence.
LQG theorem II: We should always look for quantization Q(T) for any
interesting theory T. Q(T) always exists.
LQG corollary IIa: The most interesting generalization of quantum
mechanics is the search for a Hilbert space and operators corresponding to
a given classical phase space. The direction of progress is to consider
increasingly complicated phase spaces.
LQG corollary IIb: The most interesting parts of physics are the tools
(e.g. choice of variables) how to perform the quantization; they are more
important than the ability of the theories to predict and the results
themselves. Examples of these tools are Ashtekar's variables, spin
networks, and spin foam.
LQG corollary IIc: Unusual properties of a quantum theory - such as
non-separable Hilbert space; non-existence of a path-integral formulation;
absence of doable (e.g. perturbative) calculations (of the S-matrix) do
not constitute a problem as long as the theory can be called Q(T) - which
is the most important thing.
LQG implication III: A quantum theory is as interesting as its classical
counterpart. An interesting quantum theory can only be obtained by
quantizing a classical theory.
LQG corollary IIIa: A direct quantization of pure GR should be attempted,
despite failures, because GR is a nice and simple classical theory.
LQG corollary IV: A classical theory T has nice and transparent relations
between the observable. Its quantization Q(T) should therefore have nice
relations between the observables, too.
LQG corollary IVa: Nonzero anomalies are impossible. This includes the
central extensions of the algebra, such as the central charge of a CFT.
One should assume that they are zero, and only look for a proof
(representation) of this assumption.
LQG corollary IVb: Ultraviolet divergences are impossible, too.
LQG corollary IVc: The non-renormalizable divergences can only appear if
the quantization Q(T) is done incorrectly.
LQG corollary IVd (optional): The uncertainty principle must be incorrect
because it destroys the nice relations between the observables.
OK, let's stop with the LQG crap now.
Reality I. A quantum theory is the most radical transformation of the
concepts of physics that we have seen so far, and it implies a lot of new
and unexpected phenomena. The classical intuition is often incorrect in
the quantum world, and no one-to-one correspondence exists. Quantum
mechanics defines new rules which theory are predictive, interesting, and
consistent - rules whose majority is not contained in the classical limit.
Reality II. A classical theory often does not have a consistent quantum
counterpart. An attempt to quantize a theory is often spoiled by anomalies
or non-renormalizable divergences. There are many interesting classical
theories (such as GR of Fermi's theory) that cannot be quantized. There
are also many interesting quantum theories whose classical counterparts
are not too interesting, for example because they are trivial
(Chern-Simons theory). "Reality III" below offers more examples.
Reality IIa. General quantum mechanical systems with a Hamiltonian that is
an arbitrary function on the phase space - which itself has an arbitrary
shape and topology - have not been too important and interesting in
observable physics. Virtually every interesting application of quantum
mechanics of point-like particles is based on wavefunctions on a
well-defined configuration space (x); the spectrum of the momentum
operator is therefore easily calculable. The really interesting
ramifications of quantum mechanics involve different ideas than mixing of
the coordinates and the momenta and making the geometry of the space space
a bit more complicated. In useful applications, the coordinates and
momenta are kinematically decoupled, and one can view this fact as a
non-rigorous, simple version of a Coleman-Mandula-like theorem.
The phase spaces of complicated topology are not too revolutionary from
another reason: if they are small (small number of quantum states), the
classical-quantum link is too fuzzy and ambiguous. If they are large, on
the other hand, the "local" physics on the phase space reproduces physics
on infinite smooth phase space, and is therefore qualitatively understood.
The progress in physics in the last 70 years has been based on very
different ideas than convoluted phase spaces, especially the observed new
ways how relatively simple, "polynomial" systems of very well-defined
degrees of freedom can behave, and new local symmetries and their origin.
This includes gauge theories; Higgs mechanism; confinement; insights of
the renormalization group; generation of spacetime physics from the
stringy worldsheet, and so on.
Reality IIb. In a consistent quantum theory with a classical limit, the
different procedures of quantization must be equivalent. A redefinition of
variables can help us to understand some phenomena, but it cannot make an
inconsistent theory consistent. Finally, methods are only interesting if
they lead to interesting results, and a structure claimed to be Q(T) is
not necessarily an interesting result.
Reality IIc. A physically useful quantum theory must have a separable
Hilbert space - or at least the physically relevant sector must be
separable. A theory must be able to make some predictions that can be
ruled out, at least in principle. Whether or not a system of ideas can be
called Q(T) for some classical theory T is not important physically. The
non-existence of the path-integral formulation or another usual approach
to the theory indicates that the theory has some internal problems.
Reality III. The only success story in quantizing a known classical theory
above 1 dimension was QED. Electrodynamics was a classically studied
theory that happened to have a perturbatively consistent quantum
extension. No other fundamental force is an example. Pure GR is
inconsistent as a quantum theory. The weak sector of the electroweak
theory was found directly as a quantum theory; the virtual (quantum)
W-bosons and the Z-bosons were proposed before the classical limit of the
electroweak theory (and physical, external gauge bosons, together with
their vevs) were considered: another example in which the interesting
quantum theory had to be found before its classical counterpart that is
not so interesting. Also, the equations of QCD were seriously proposed
only when it was known that the (quantum) \beta function was negative and
the force was able to confine quarks and describe asymptotic freedom at
high energies. The classical QCD was not an interesting theory either
because it led to new, unobserved long-range forces, as Pauli proved
(neglecting quantum mechanics in the field-theoretical context) in the
1950s.
The question whether a quantum theory is interesting is largerly
uncorrelated with the question whether its classical limit is interesting.
Examples have been given in "Reality IIa". More generally, a quantum
theory is only interesting if it is predictive. It is very important for
the predictive power of a theory that a generic (high-order) interaction
would lead to inconsistencies in the UV. If it did not, infinitely many
coefficients would be undetermined. In consistent theories they can either
be neglected, or their coefficients' smallness can be justified because
these are the irrelevant operators associated with new physics at much
higher scales, as the logic of the renormalization group dictates, and the
super-high energy physics does not directly influence physics at
accessible energies. A framework that would allow arbitrary couplings to
be added would imply a complete loss of predictive power. String theory is
the only known framework that constrains the parameters in a quantum
theory by mechanisms that go beyond the RG.
These are the physically relevant questions, the "landscape" of
interesting field theories is mapped by the Renormalization Group (RG),
and these tools have nothing to do with the beauty of the classical limit.
Incidentally, the beauty of a classical theory cannot be directly
connected with predictivity - because new, more complicated terms can
essentially always be added - and therefore the "beauty" of a classical
theory is rather close to the subjective notion of simplicity whose direct
physical importance is very limited: a classical GR with R^2 terms is not
inferior to the pure GR, except for purely subjective reasons. On the
other hand, quantum physics of gravity gives a principle to label R^2 as
the less important term.
On the other hand, it is also believed that some interesting quantum
theories do not follow from a classical starting point. It is also known
for sure that a quantum theory can have several classical limits.
Reality IIIa: GR is an example of a non-renormalizable theory, and the
problematic UV behavior is a key to new physics that must regularize the
divergences. In the case of Fermi's theory, the UV completion turns out to
be essentially unique - a spontaneously broken gauge theory. General
covariance is as much constraining or more, and a UV completion must be
taken seriously. String theory is the only known solution how to generate
consistent loop (quantum) contributions to the scattering amplitudes.
The only special feature of GR is the general covariance, which is a
generalization of gauge symmetries (with a spin greater by one). The
structure respecting this gauge symmetry is important for classical
decoupling of the unphysical modes - at the non-linear, multiparticle
level - but it does not imply anything about the quantum loops of GR.
Reality IV: Quantum field theory implies that the observables do have
short distance divergences in their products. Algebras can acquire new
terms that did not exist classically. This behavior can be measured in
some cases; in other cases it directly follows from a nonzero commutator
(the uncertainty principle) and the equations of motion.
Reality IVa: Chiral gauge theories and similar theories in 2k dimensions
typically lead to gauge anomalies. Gauge anomalies imply that the theory
is no longer consistent because the unphysical (and negative-norm)
polarizations of the gauge bosons (or graviton) no longer decouple. The
structure of anomalies can be calculated if one tries to regularize
certain UV divergences, but the result is purely reflected by the low
energy (IR) spectrum of the theory. There is no way to avoid the
conclusion about gauge anomalies except for cancelling them.
Anomalies in global symmetries do not imply an inconsistency, but they
drastically influence the physics of a given theory.
Central charge (or Weyl anomaly) is a specific example of an anomalous
term that can be calculated in many ways, and there is no consistent way
to avoid its nonzero value.
Reality IVb: The existence of UV divergences is real. This existence is in
fact essential to our ability to set the coupling constants and other
parameters equal to their observed values.
Reality IVc: Non-renormalizability of some UV divergences is not a
speculation, but a result of quantitative calculations. The only way how
these problems could disappear in a reformulated theory is the situation
that the given theory has a UV fixed point under the renormalization
group. The non-trivial, interacting fixed points are very rare, and there
exists no good reason why such a fixed point should exist and be
associated with a generic non-renormalizable theory.
Reality IVd (optional): The uncertainty principle is the basic principle
behind the whole structure of quantum physics and everything that contains
the adjective "quantum".
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E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
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