gralp@poczta.onet.pl
Aug12-04, 08:30 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\n\n\n> jack (jfisher@mn.rr.com) wrote:\n>\n> I appologize if this topic has been delt with here before,though I\n> cannot find a satisfactory elementary definition in the google\n> archives.\n> What does it mean to say a field has " an infinite degrees of\n> freedom"?\n> I will include a small passage from A.O. Barut\'s " Electrodynamics\n> and classical theory of fields and particles" to start.\n> " The infinite number of degrees of fredom of the field must be\n> described by continuous indices. Instead of the coordiates q_1,q_2 ...\n> ,the dynamicalvariables of the field will be a set of functions\n> psi^a(X,t), a=1,.....N where (X,t) are now parameter which,together\n> with a,label the degrees of freedom of the system."\n> Now in classical mechanics I understand that the number of\n> variables in the Lagrangian of the system is not synonomous with the\n> number of degrees of freedom because one can come up with different\n> configuation space variables. The electromagnetic field in vacuum can\n> be described by either six functions(three of E and 3 of B) or 4\n> function A^u which are not unique. The free electromagnetic field may\n> be decribed by several different Lagrangians all with a finite number\n> of field variables e.g. E and B, or A^u and by a finite number of\n> field equations. So where does the "infinite number of degrees of\n> freedom com from? Cann someone deliniate exactly what the\n> difference is in the definition of "degrees of freedom" as it pertains\n> to the classical point particles formulation and classical field theory?\n\nActualy the counting of degrees of freedom (DOFs) is - at least in my understanding - a bit subtle and context-dependent.\n\nBegin with a state space H, i.e. the space of exclusive propositions, and assume some distance measure on it. If for each state one is able to find an arbitrarily close neighbour, then one is safe to assume continuity of DOFs and the state space cardinali\nty of reals |H|=|R|. At the very moment of conjuring such an abstract, one tacitly assumes that there is a way to address each particular state psi in H with a label psi^A. If H is finite, so are the labels, otherwise they may yet be countable or not. In\na "classical" description the exclusiveness of states means, they are not regarded as independent DOFs, but in probabilistic or quantum approach one relaxes this constraint allowing for mixtures and superpositions respectively, therefore making it appropr\niate to call each state by DOF.\n\nIt is all too often desirable to further replace the general labeling with a factorized form psi^{A(x,y,...)}, by distinguishing several "dimensions" or "properties" which can be equivalently used for the purpose. There is of course a lot of arbitrarines\nin such a decomposition, and we seek the dimensions {x,y...} to be orthogonal, but in principle this may not be so - any surjection\n\n{x,y...} -> A\n\nis equally valid. These dimensions are also called "degrees of freedom" thus leading to somewhat confusing situation when a state with infinite DOFs, psi^A, can be equivalently represented by several DOFs, for instance\n\npsi^{A(a,x,t)} = psi^a(x,t), a in N, x,t in R.\n\nIn one sense there are finite number of DOFs here (3), in the other - they "unfold" to an uncountable and therefore infinite number of DOFs/states in H. Whether one calls by DOFs the cardinality of the state space |H|, or the number of properties used to\nparametrize the states is a matter of convention and depend on the purpose of the approach. In classical dynamics it is customary to identify DOFs with the spatial (or spacetime) dimensions, while in field theory - they are the labels of available states.\n\n\n\nregards\npg\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>jack (jfisher@mn.rr.com) wrote:
>
> I appologize if this topic has been delt with here before,though I
> cannot find a satisfactory elementary definition in the google
> archives.
> What does it mean to say a field has " an infinite degrees of
> freedom"?
> I will include a small passage from A.O. Barut's " Electrodynamics
> and classical theory of fields and particles" to start.
> " The infinite number of degrees of fredom of the field must be
> described by continuous indices. Instead of the coordiates q_1,q_2 ...
> ,the dynamicalvariables of the field will be a set of functions
> \psi^a(X,t), a=1,.....N where (X,t) are now parameter which,together
> with a,label the degrees of freedom of the system."
> Now in classical mechanics I understand that the number of
> variables in the Lagrangian of the system is not synonomous with the
> number of degrees of freedom because one can come up with different
> configuation space variables. The electromagnetic field in vacuum can
> be described by either six functions(three of E and 3 of B) or 4
> function A^u which are not unique. The free electromagnetic field may
> be decribed by several different Lagrangians all with a finite number
> of field variables e.g. E and B, or A^u and by a finite number of
> field equations. So where does the "infinite number of degrees of
> freedom com from? Cann someone deliniate exactly what the
> difference is in the definition of "degrees of freedom" as it pertains
> to the classical point particles formulation and classical field theory?
Actualy the counting of degrees of freedom (DOFs) is - at least in my understanding - a bit subtle and context-dependent.
Begin with a state space H, i.e. the space of exclusive propositions, and assume some distance measure on it. If for each state one is able to find an arbitrarily close neighbour, then one is safe to assume continuity of DOFs and the state space cardinali
ty of reals |H|=|R|. At the very moment of conjuring such an abstract, one tacitly assumes that there is a way to address each particular state \psi in H with a label \psi^A. If H is finite, so are the labels, otherwise they may yet be countable or not. In
a "classical" description the exclusiveness of states means, they are not regarded as independent DOFs, but in probabilistic or quantum approach one relaxes this constraint allowing for mixtures and superpositions respectively, therefore making it appropr
iate to call each state by DOF.
It is all too often desirable to further replace the general labeling with a factorized form \psi^{A(x,y,...)}, by distinguishing several "dimensions" or "properties" which can be equivalently used for the purpose. There is of course a lot of arbitrarines
in such a decomposition, and we seek the dimensions {x,y...} to be orthogonal, but in principle this may not be so - any surjection
{x,y[/itex]...[itex]} -> A
is equally valid. These dimensions are also called "degrees of freedom" thus leading to somewhat confusing situation when a state with infinite DOFs, \psi^A, can be equivalently represented by several DOFs, for instance
\psi^{A(a,x,t)} = \psi^a(x,t), a in N, x,t in R.
In one sense there are finite number of DOFs here (3), in the other - they "unfold" to an uncountable and therefore infinite number of DOFs/states in H. Whether one calls by DOFs the cardinality of the state space |H|, or the number of properties used to
parametrize the states is a matter of convention and depend on the purpose of the approach. In classical dynamics it is customary to identify DOFs with the spatial (or spacetime) dimensions, while in field theory - they are the labels of available states.
regards
pg
>
> I appologize if this topic has been delt with here before,though I
> cannot find a satisfactory elementary definition in the google
> archives.
> What does it mean to say a field has " an infinite degrees of
> freedom"?
> I will include a small passage from A.O. Barut's " Electrodynamics
> and classical theory of fields and particles" to start.
> " The infinite number of degrees of fredom of the field must be
> described by continuous indices. Instead of the coordiates q_1,q_2 ...
> ,the dynamicalvariables of the field will be a set of functions
> \psi^a(X,t), a=1,.....N where (X,t) are now parameter which,together
> with a,label the degrees of freedom of the system."
> Now in classical mechanics I understand that the number of
> variables in the Lagrangian of the system is not synonomous with the
> number of degrees of freedom because one can come up with different
> configuation space variables. The electromagnetic field in vacuum can
> be described by either six functions(three of E and 3 of B) or 4
> function A^u which are not unique. The free electromagnetic field may
> be decribed by several different Lagrangians all with a finite number
> of field variables e.g. E and B, or A^u and by a finite number of
> field equations. So where does the "infinite number of degrees of
> freedom com from? Cann someone deliniate exactly what the
> difference is in the definition of "degrees of freedom" as it pertains
> to the classical point particles formulation and classical field theory?
Actualy the counting of degrees of freedom (DOFs) is - at least in my understanding - a bit subtle and context-dependent.
Begin with a state space H, i.e. the space of exclusive propositions, and assume some distance measure on it. If for each state one is able to find an arbitrarily close neighbour, then one is safe to assume continuity of DOFs and the state space cardinali
ty of reals |H|=|R|. At the very moment of conjuring such an abstract, one tacitly assumes that there is a way to address each particular state \psi in H with a label \psi^A. If H is finite, so are the labels, otherwise they may yet be countable or not. In
a "classical" description the exclusiveness of states means, they are not regarded as independent DOFs, but in probabilistic or quantum approach one relaxes this constraint allowing for mixtures and superpositions respectively, therefore making it appropr
iate to call each state by DOF.
It is all too often desirable to further replace the general labeling with a factorized form \psi^{A(x,y,...)}, by distinguishing several "dimensions" or "properties" which can be equivalently used for the purpose. There is of course a lot of arbitrarines
in such a decomposition, and we seek the dimensions {x,y...} to be orthogonal, but in principle this may not be so - any surjection
{x,y[/itex]...[itex]} -> A
is equally valid. These dimensions are also called "degrees of freedom" thus leading to somewhat confusing situation when a state with infinite DOFs, \psi^A, can be equivalently represented by several DOFs, for instance
\psi^{A(a,x,t)} = \psi^a(x,t), a in N, x,t in R.
In one sense there are finite number of DOFs here (3), in the other - they "unfold" to an uncountable and therefore infinite number of DOFs/states in H. Whether one calls by DOFs the cardinality of the state space |H|, or the number of properties used to
parametrize the states is a matter of convention and depend on the purpose of the approach. In classical dynamics it is customary to identify DOFs with the spatial (or spacetime) dimensions, while in field theory - they are the labels of available states.
regards
pg