Lubos Motl
Apr22-04, 09:02 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>On Thu, 22 Apr 2004, Karim wrote:\n\n> Are strings subject to the same principles that point particles are\n> in quantum mechanics, i.e., is each point on a string associated with\n> an abstract state vector and observables obeying uncertainty relations?\n\nThis seems as a very correct description. Strings, as studied by string\ntheorists, are subject to quantum mechanics, much like pointlike\nparticles. For example, a classical string would tend to shrink to zero\nsize and stop any vibrations to minimize its energy. The real, quantum\nstring can\'t do it because of the uncertainty principle: it must make a\ncompromise between the position uncertainty and momentum uncertainty.\n\nIt wants to minimize its vibrations - the momentum p(sigma) at each point\nsigma (coordinate along string) - because it contributes to the kinetic\nenergy. However it also wants to minimize x\'(sigma), the derivative of the\ncoordinate along the string, because it contributes to the length of the\nstring, which also costs some energy (tension times length). But p(sigma)\nand x(sigma) can\'t be simultaneously zero, and therefore the string finds\nan appropriate compromise. This compromise is the ground state |0>, and it\ncan be described as infinitely many harmonic oscillators in their ground\nstates. The higher-energetic excitations are obtained by acting with the\ncreation operators on this state |0> - there is one creation oscillator\nfor each choice of the numbers (mu,n) where "mu" is the transverse\ndirection in space (1...24 in bosonic string theory), and n=1,2,3... is\nthe Fourier mode of this coordinate along the string.\n\nIn bosonic string theory, the old-fashioned string theory in 26 dimensions\nwithout any fermions, the ground state |0> is the tachyon, and the\ngraviton is obtained as a_{-1}^mu a~_{-1}^nu |0> where a,a~ are certain\ncreation operators; note that the state has two spacetime vector indices\nmu,nu, which is OK for the metric (graviton). Much like in the harmonic\noscillator where these creation operators raise the energy, the creation\noperators of a string raise the squared mass of the particle that\napproximates the vibrating string. The squared mass of the ground state\ntachyon |0> is negative - in some sense, it is proportional to the sum\n1+2+3+4+5+... which is in some deep sense equal to -1/12 (which is\nmultiplied by the analogue of "hbar.omega/2") - and the mass of the\ngraviton is zero.\n\nFor point-like particles you have the commutation relations [x,p]=i.hbar,\nand for strings you have analogously\n\n[x(sigma),p(sigma\')] = i.hbar.delta(sigma-sigma\')\n\nwhere delta is Dirac\'s distribution - a function that vanishes unless\nsigma=sigma\'. Sigma and sigma\' are the coordinate along the string.\n\n> That is, if I choose to picture an electron as a "fuzzy" point according to\n> quantum mechanics, should I picture an electron as a "fuzzy" string\n> (in how many ever dimensions) according to string theory?\n\nDefinitely. Such a real string has some probability amplitude to "be" in\nthe state associated with some shape, and these "wave functionals" are\nanalogous to the "clouds" of wavefunctions for the electron in the\nHydrogen atom. The difference is that strings have many more degrees of\nfreedom - one must study the behavior of x,p at each point sigma of the\nstring - and therefore strings are a bit more complicated than the\nelectrons. However they lead to the infinite family of harmonic\noscillators mentioned above - and harmonic oscillators are in some sense\n*easier* than the Hydrogen atom.\n\nCheers,\nLubos\n_________________________ __________________________________________________ ___\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 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>On Thu, 22 Apr 2004, Karim wrote:
> Are strings subject to the same principles that point particles are
> in quantum mechanics, i.e., is each point on a string associated with
> an abstract state vector and observables obeying uncertainty relations?
This seems as a very correct description. Strings, as studied by string
theorists, are subject to quantum mechanics, much like pointlike
particles. For example, a classical string would tend to shrink to zero
size and stop any vibrations to minimize its energy. The real, quantum
string can't do it because of the uncertainty principle: it must make a
compromise between the position uncertainty and momentum uncertainty.
It wants to minimize its vibrations - the momentum p(\sigma) at each point
\sigma (coordinate along string) - because it contributes to the kinetic
energy. However it also wants to minimize x'(\sigma), the derivative of the
coordinate along the string, because it contributes to the length of the
string, which also costs some energy (tension times length). But p(\sigma)
and x(\sigma) can't be simultaneously zero, and therefore the string finds
an appropriate compromise. This compromise is the ground state |0>, and it
can be described as infinitely many harmonic oscillators in their ground
states. The higher-energetic excitations are obtained by acting with the
creation operators on this state |0> - there is one creation oscillator
for each choice of the numbers (\mu,n) where "\mu" is the transverse
direction in space (1...24 in bosonic string theory), and n=1,2,3... is
the Fourier mode of this coordinate along the string.
In bosonic string theory, the old-fashioned string theory in 26 dimensions
without any fermions, the ground state |0> is the tachyon, and the
graviton is obtained as a_{-1}^\mu a~_{-1}^\nu |0> where a,a~ are certain
creation operators; note that the state has two spacetime vector indices
\mu,\nu, which is OK for the metric (graviton). Much like in the harmonic
oscillator where these creation operators raise the energy, the creation
operators of a string raise the squared mass of the particle that
approximates the vibrating string. The squared mass of the ground state
tachyon |0> is negative - in some sense, it is proportional to the sum
1+2+3+4+5+... which is in some deep sense equal to -1/12 (which is
multiplied by the analogue of "\hbar.\omega/2") - and the mass of the
graviton is zero.
For point-like particles you have the commutation relations [x,p]=i.\hbar,
and for strings you have analogously
[x(\sigma),p(\sigma')] = i[/itex].\hbar.[itex]\delta(\sigma-\sigma')
where \delta is Dirac's distribution - a function that vanishes unless
\sigma=\sigma'. \Sigma and \sigma' are the coordinate along the string.
> That is, if I choose to picture an electron as a "fuzzy" point according to
> quantum mechanics, should I picture an electron as a "fuzzy" string
> (in how many ever dimensions) according to string theory?
Definitely. Such a real string has some probability amplitude to "be" in
the state associated with some shape, and these "wave functionals" are
analogous to the "clouds" of wavefunctions for the electron in the
Hydrogen atom. The difference is that strings have many more degrees of
freedom - one must study the behavior of x,p at each point \sigma of the
string - and therefore strings are a bit more complicated than the
electrons. However they lead to the infinite family of harmonic
oscillators mentioned above - and harmonic oscillators are in some sense
*easier* than the Hydrogen atom.
Cheers,
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> Are strings subject to the same principles that point particles are
> in quantum mechanics, i.e., is each point on a string associated with
> an abstract state vector and observables obeying uncertainty relations?
This seems as a very correct description. Strings, as studied by string
theorists, are subject to quantum mechanics, much like pointlike
particles. For example, a classical string would tend to shrink to zero
size and stop any vibrations to minimize its energy. The real, quantum
string can't do it because of the uncertainty principle: it must make a
compromise between the position uncertainty and momentum uncertainty.
It wants to minimize its vibrations - the momentum p(\sigma) at each point
\sigma (coordinate along string) - because it contributes to the kinetic
energy. However it also wants to minimize x'(\sigma), the derivative of the
coordinate along the string, because it contributes to the length of the
string, which also costs some energy (tension times length). But p(\sigma)
and x(\sigma) can't be simultaneously zero, and therefore the string finds
an appropriate compromise. This compromise is the ground state |0>, and it
can be described as infinitely many harmonic oscillators in their ground
states. The higher-energetic excitations are obtained by acting with the
creation operators on this state |0> - there is one creation oscillator
for each choice of the numbers (\mu,n) where "\mu" is the transverse
direction in space (1...24 in bosonic string theory), and n=1,2,3... is
the Fourier mode of this coordinate along the string.
In bosonic string theory, the old-fashioned string theory in 26 dimensions
without any fermions, the ground state |0> is the tachyon, and the
graviton is obtained as a_{-1}^\mu a~_{-1}^\nu |0> where a,a~ are certain
creation operators; note that the state has two spacetime vector indices
\mu,\nu, which is OK for the metric (graviton). Much like in the harmonic
oscillator where these creation operators raise the energy, the creation
operators of a string raise the squared mass of the particle that
approximates the vibrating string. The squared mass of the ground state
tachyon |0> is negative - in some sense, it is proportional to the sum
1+2+3+4+5+... which is in some deep sense equal to -1/12 (which is
multiplied by the analogue of "\hbar.\omega/2") - and the mass of the
graviton is zero.
For point-like particles you have the commutation relations [x,p]=i.\hbar,
and for strings you have analogously
[x(\sigma),p(\sigma')] = i[/itex].\hbar.[itex]\delta(\sigma-\sigma')
where \delta is Dirac's distribution - a function that vanishes unless
\sigma=\sigma'. \Sigma and \sigma' are the coordinate along the string.
> That is, if I choose to picture an electron as a "fuzzy" point according to
> quantum mechanics, should I picture an electron as a "fuzzy" string
> (in how many ever dimensions) according to string theory?
Definitely. Such a real string has some probability amplitude to "be" in
the state associated with some shape, and these "wave functionals" are
analogous to the "clouds" of wavefunctions for the electron in the
Hydrogen atom. The difference is that strings have many more degrees of
freedom - one must study the behavior of x,p at each point \sigma of the
string - and therefore strings are a bit more complicated than the
electrons. However they lead to the infinite family of harmonic
oscillators mentioned above - and harmonic oscillators are in some sense
*easier* than the Hydrogen atom.
Cheers,
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^