why do complex numbers work in physics? [Archive]

alistair

Dec3-04, 04:51 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>The square root of -1 is used a lot in physics.\nBut how does it relate to what,I suspect,most people would regard\nas the real world i.e real numbers (for example we speak of real\nprobabilities\nand not imaginary probabilities - real probabilities are the "real"\nworld).\nComplex numbers can be represented by two orthogonal axes on a sheet\nof paper and so can real numbers.Since such representations are both\ngeometrical\nentities,do complex numbers only relate to real numbers (and hence the\n"real" world) in the context of geometry? And since general relativity\nis a theory based on ideas of geometry, do complex numbers only relate\nto the real\nworld in the context of general relativity i.e would an imaginary\nprobability seem reasonable in the theory of general relativity?\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>The square root of -1 is used a lot in physics.
But how does it relate to what,I suspect,most people would regard
as the real world i.e real numbers (for example we speak of real
probabilities
and not imaginary probabilities - real probabilities are the "real"
world).
Complex numbers can be represented by two orthogonal axes on a sheet
of paper and so can real numbers.Since such representations are both
geometrical
entities,do complex numbers only relate to real numbers (and hence the
"real" world) in the context of geometry? And since general relativity
is a theory based on ideas of geometry, do complex numbers only relate
to the real
world in the context of general relativity i.e would an imaginary
probability seem reasonable in the theory of general relativity?

Igor Khavkine

Dec4-04, 03:15 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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0412021058.4771a6ad@posting.google. com>...\n> The square root of -1 is used a lot in physics.\n> But how does it relate to what,I suspect,most people would regard\n> as the real world i.e real numbers (for example we speak of real\n> probabilities\n> and not imaginary probabilities - real probabilities are the "real"\n> world).\n\nThe real numbers are not the real world. The world cares very little\nabout what mathematics we use to describe it. Complex numbers are\nused in physics for the same reason that real numbers are: because\nit\'s\npossible. If you want an analogy, think about trying to solve a\ndifferential equation. The only way general way to do it is to guess\na solution and then check whether it works. If a certain mathematical\nstructure (e.g. complex numbers) allows you to the right guess, by all\nmeans use it. This reason is deep enough for physics. If you want to\nknow deep mathematical reasons why certain mathematical structures\ncome up in solutions of certain mathematical problems, that\'s a\ndifferent\nquestion.\n\nBTW, probability theory is not magic. It is based on common sense. If\nyou\ncan make sense of complex probabilities, feel free to construct a\nprobability theory based on them. But do not expect the converse, that\nthrowing in complex numbers into existing probability theory will\nallow you to make sense of complex probabilities.\n\n> Complex numbers can be represented by two orthogonal axes on a sheet\n> of paper and so can real numbers.Since such representations are both\n> geometrical\n> entities,do complex numbers only relate to real numbers (and hence the\n> "real" world) in the context of geometry? And since general relativity\n> is a theory based on ideas of geometry, do complex numbers only relate\n> to the real\n> world in the context of general relativity i.e would an imaginary\n> probability seem reasonable in the theory of general relativity?\n\nWow! Now that\'s a non sequitur if I\'ve ever seen one. See my comments\nabove about probability and complex numbers. In order to avoid making\nvaccuous speculative statements, you might want to familiarize\nyourself with logic and rules of inference. It would also help to\nrealize that even though physics can be described by some mathematics,\nnot all mathematics describes some physics.\n\nIgor\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0412021058.4771a6ad@posting.google.com>...
> The square root of -1 is used a lot in physics.
> But how does it relate to what,I suspect,most people would regard
> as the real world i.e real numbers (for example we speak of real
> probabilities
> and not imaginary probabilities - real probabilities are the "real"
> world).

The real numbers are not the real world. The world cares very little
about what mathematics we use to describe it. Complex numbers are
used in physics for the same reason that real numbers are: because
it's
possible. If you want an analogy, think about trying to solve a
differential equation. The only way general way to do it is to guess
a solution and then check whether it works. If a certain mathematical
structure (e.g. complex numbers) allows you to the right guess, by all
means use it. This reason is deep enough for physics. If you want to
know deep mathematical reasons why certain mathematical structures
come up in solutions of certain mathematical problems, that's a
different
question.

BTW, probability theory is not magic. It is based on common sense. If
you
can make sense of complex probabilities, feel free to construct a
probability theory based on them. But do not expect the converse, that
throwing in complex numbers into existing probability theory will
allow you to make sense of complex probabilities.

> Complex numbers can be represented by two orthogonal axes on a sheet
> of paper and so can real numbers.Since such representations are both
> geometrical
> entities,do complex numbers only relate to real numbers (and hence the
> "real" world) in the context of geometry? And since general relativity
> is a theory based on ideas of geometry, do complex numbers only relate
> to the real
> world in the context of general relativity i.e would an imaginary
> probability seem reasonable in the theory of general relativity?

Wow! Now that's a non sequitur if I've ever seen one. See my comments
above about probability and complex numbers. In order to avoid making
vaccuous speculative statements, you might want to familiarize
yourself with logic and rules of inference. It would also help to
realize that even though physics can be described by some mathematics,
not all mathematics describes some physics.

Igor

Patrick Powers

Dec4-04, 03:16 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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0412021058.4771a6ad@posting.google. com>...\n> The square root of -1 is used a lot in physics.\n> But how does it relate to what,I suspect,most people would regard\n> as the real world i.e real numbers (for example we speak of real\n> probabilities\n> and not imaginary probabilities - real probabilities are the "real"\n> world).\n> Complex numbers can be represented by two orthogonal axes on a sheet\n> of paper and so can real numbers.Since such representations are both\n> geometrical\n> entities,do complex numbers only relate to real numbers (and hence the\n> "real" world) in the context of geometry? And since general relativity\n> is a theory based on ideas of geometry, do complex numbers only relate\n> to the real\n> world in the context of general relativity i.e would an imaginary\n> probability seem reasonable in the theory of general relativity?\n\nComplex numbers are used to represent cyclic processes, particularly\nthose that involve sines and cosines. This includes electromagnetic\nwaves, so it is very important in physics. They also are used for the\nFourier transform, which is one of the most fundamental\nrepresentations of reality. You COULD do all these computations with\nsines and cosines and real numbers, but it would be so awkward it\nwould make no sense.\n\nEvery complex polynomial has a solution, so in this sense complex\nnumbers are easier to work with. The real numbers is a subset of the\ncomplex numbers, so sometimes one can work with the complex numbers\nthen just use the real solution, if there is one. So one can think of\nthe complex numbers as more general and basic than the reals.\n\nBy the way, almost all of the real numbers are quite imaginary and\ntheoretical, not real at all. And that "square root of -1" stuff\nreally doesn\'t have all that much to do with it. It is mostly\nhistorical: that is the way the complex numbers got started, but it\ndoesn\'t have much meaning. It is better to look at the complex\nnumbers as a whole new system of computation with very useful\nproperties. This is very common in mathematics: sometimes the easiest\nway to do something is make a whole new system of computation that is\na superset of some other system.\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0412021058.4771a6ad@posting.google.com>...
> The square root of -1 is used a lot in physics.
> But how does it relate to what,I suspect,most people would regard
> as the real world i.e real numbers (for example we speak of real
> probabilities
> and not imaginary probabilities - real probabilities are the "real"
> world).
> Complex numbers can be represented by two orthogonal axes on a sheet
> of paper and so can real numbers.Since such representations are both
> geometrical
> entities,do complex numbers only relate to real numbers (and hence the
> "real" world) in the context of geometry? And since general relativity
> is a theory based on ideas of geometry, do complex numbers only relate
> to the real
> world in the context of general relativity i.e would an imaginary
> probability seem reasonable in the theory of general relativity?

Complex numbers are used to represent cyclic processes, particularly
those that involve sines and cosines. This includes electromagnetic
waves, so it is very important in physics. They also are used for the
Fourier transform, which is one of the most fundamental
representations of reality. You COULD do all these computations with
sines and cosines and real numbers, but it would be so awkward it
would make no sense.

Every complex polynomial has a solution, so in this sense complex
numbers are easier to work with. The real numbers is a subset of the
complex numbers, so sometimes one can work with the complex numbers
then just use the real solution, if there is one. So one can think of
the complex numbers as more general and basic than the reals.

By the way, almost all of the real numbers are quite imaginary and
theoretical, not real at all. And that "square root of -1" stuff
really doesn't have all that much to do with it. It is mostly
historical: that is the way the complex numbers got started, but it
doesn't have much meaning. It is better to look at the complex
numbers as a whole new system of computation with very useful
properties. This is very common in mathematics: sometimes the easiest
way to do something is make a whole new system of computation that is
a superset of some other system.

bluechip

Dec4-04, 03:17 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>alistair wrote:\n> The square root of -1 is used a lot in physics.\n> But how does it relate to what,I suspect,most people would regard\n> as the real world i.e real numbers (for example we speak of real\n> probabilities\n> and not imaginary probabilities - real probabilities are the "real"\n> world).\n\nThis is an example of where one\'s common sense can let one down.\nConsider a field (the type with grass) in which there are several sheep.\nA farmer standing at the gate of said field could quite easily count his\nsheep thus: "One, two, three, four, ..." However, apart from sounding\nslightly bizarre, there is nothing to stop him from counting them like\nthis: "One plus zero i, two plus zero i, three plus zero i, ..." The\npoint is that the set of real numbers is a well defined subset of the\nset of complex numbers (it is simply the set of all complex numbers a+bi\nfor a,b in R and b=0).\n\nThe moral of the story? Just because one uses real numbers to analyse\ncertain physical problems, one shouldn\'t lose sight of the fact that one\nis dealing with *complex* numbers all the time.\n\n> Complex numbers can be represented by two orthogonal axes on a sheet\n> of paper and so can real numbers.\n\nComplex numbers can be represented as *points* in such a space, where\none axis denotes the real component and the other the complex component.\nThe union of all such points is then the space of complex numbers.\n\n> Since such representations are both\n> geometrical\n> entities,do complex numbers only relate to real numbers (and hence the\n> "real" world) in the context of geometry?\n\nAs said earlier, the set R^n is a subset of C^n - that is how they are\nrelated. The interesting thing, of course, is the special mapping from\nC^n->R^n that you should be familiar with...\n\n> And since general relativity\n> is a theory based on ideas of geometry, do complex numbers only relate\n> to the real world in the context of general relativity i.e would an imaginary\n> probability seem reasonable in the theory of general relativity?\n>\n\nThis sentence doesn\'t make any sense.\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>alistair wrote:
> The square root of -1 is used a lot in physics.
> But how does it relate to what,I suspect,most people would regard
> as the real world i.e real numbers (for example we speak of real
> probabilities
> and not imaginary probabilities - real probabilities are the "real"
> world).

This is an example of where one's common sense can let one down.
Consider a field (the type with grass) in which there are several sheep.
A farmer standing at the gate of said field could quite easily count his
sheep thus: "One, two, three, four, ..." However, apart from sounding
slightly bizarre, there is nothing to stop him from counting them like
this: "One plus zero i, two plus zero i, three plus zero i, ..." The
point is that the set of real numbers is a well defined subset of the
set of complex numbers (it is simply the set of all complex numbers a+bi
for a,b in R and b=0).

The moral of the story? Just because one uses real numbers to analyse
certain physical problems, one shouldn't lose sight of the fact that one
is dealing with *complex* numbers all the time.

> Complex numbers can be represented by two orthogonal axes on a sheet
> of paper and so can real numbers.

Complex numbers can be represented as *points* in such a space, where
one axis denotes the real component and the other the complex component.
The union of all such points is then the space of complex numbers.

> Since such representations are both
> geometrical
> entities,do complex numbers only relate to real numbers (and hence the
> "real" world) in the context of geometry?

As said earlier, the set R^n is a subset of C^n - that is how they are
related. The interesting thing, of course, is the special mapping from
C^n->R^n that you should be familiar with...

> And since general relativity
> is a theory based on ideas of geometry, do complex numbers only relate
> to the real world in the context of general relativity i.e would an imaginary
> probability seem reasonable in the theory of general relativity?
>

This sentence doesn't make any sense.

Arnold Neumaier

Dec4-04, 03:17 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>alistair wrote:\n> The square root of -1 is used a lot in physics.\n> But how does it relate to what,I suspect,most people would regard\n> as the real world i.e real numbers (for example we speak of real\n> probabilities\n> and not imaginary probabilities - real probabilities are the "real"\n> world).\n> Complex numbers can be represented by two orthogonal axes on a sheet\n> of paper and so can real numbers.Since such representations are both\n> geometrical\n> entities,do complex numbers only relate to real numbers (and hence the\n> "real" world) in the context of geometry? And since general relativity\n> is a theory based on ideas of geometry, do complex numbers only relate\n> to the real\n> world in the context of general relativity i.e would an imaginary\n> probability seem reasonable in the theory of general relativity?\n>\nProbablities must be real numbers between 0 and 1 to have a statistical\ninterpretation; and without one they are pure nonsense.\n\nFor your remaining questions, the entry\n\'\'Why use complex numbers in physics?\'\'\nin my theoretical physics FAQ at\nhttp://www.mat.univie.ac.at/~neum/physics-faq.txt\nmight be helpful.\n\n\nArnold Neumaier\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>alistair wrote:
> The square root of -1 is used a lot in physics.
> But how does it relate to what,I suspect,most people would regard
> as the real world i.e real numbers (for example we speak of real
> probabilities
> and not imaginary probabilities - real probabilities are the "real"
> world).
> Complex numbers can be represented by two orthogonal axes on a sheet
> of paper and so can real numbers.Since such representations are both
> geometrical
> entities,do complex numbers only relate to real numbers (and hence the
> "real" world) in the context of geometry? And since general relativity
> is a theory based on ideas of geometry, do complex numbers only relate
> to the real
> world in the context of general relativity i.e would an imaginary
> probability seem reasonable in the theory of general relativity?
>
Probablities must be real numbers between and 1 to have a statistical
interpretation; and without one they are pure nonsense.

For your remaining questions, the entry
''Why use complex numbers in physics?''
in my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
might be helpful.

Arnold Neumaier

P. Gralewicz

Dec6-04, 07:17 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>bluechip <bluechip@spam.net> wrote in message news:<coshaa\\$jpj\\$1@kermit.esat.net>...\n> [...] Just because one uses real numbers to analyse\n> certain physical problems, one shouldn\'t lose sight of the fact that one\n> is dealing with *complex* numbers all the time.\n>\n> [...]\n>\n> As said earlier, the set R^n is a subset of C^n - that is how they are\n> related. The interesting thing, of course, is the special mapping from\n> C^n->R^n that you should be familiar with...\n\nThis is a biased point of view. Complex numbers are equivalent to\ndoubled reals plus a symmetry given by the complex structure. In this\nsense they are "constrained reals". Despite this, many feel justified\ncalling complex numbers superior to reals, as their in-built symmetry\nunderlies so much of a wonderful mathematics.\n\nIgor Khavkine wrote:\n> If you can make sense of complex probabilities, feel free to\n> construct a probability theory based on them.\n\nIndeed, one can make quite a good sense of complex probabilities.\n\nFor example, take a invese stereographic projection back to a\nunit-diameter sphere in such a way that the poles correspond to 0 and\n1 on the complex plane. Then compactify. The arcs passing through 0\nand 1 on the plane are mapped to great circles on the sphere, and\nre(z)=1/2 - to the equator. What one gets can be interpreted as the\nBloh sphere of a qubit - the elementary construct in quantum\ninformation theory. Ignore the phase and we are back at ordinary\nprobability (a projection from entire C-plane to the [0,1] segment, or\nany arc anchored at 0 and 1).\n\nbest\npg\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>bluechip <bluechip@spam.net> wrote in message news:<coshaa$jpj$1@kermit.esat.net>...
> [...] Just because one uses real numbers to analyse
> certain physical problems, one shouldn't lose sight of the fact that one
> is dealing with *complex* numbers all the time.
>
> [...]
>
> As said earlier, the set R^n is a subset of C^n - that is how they are
> related. The interesting thing, of course, is the special mapping from
> C^n->R^n that you should be familiar with...

This is a biased point of view. Complex numbers are equivalent to
doubled reals plus a symmetry given by the complex structure. In this
sense they are "constrained reals". Despite this, many feel justified
calling complex numbers superior to reals, as their in-built symmetry
underlies so much of a wonderful mathematics.

Igor Khavkine wrote:
> If you can make sense of complex probabilities, feel free to
> construct a probability theory based on them.

Indeed, one can make quite a good sense of complex probabilities.

For example, take a invese stereographic projection back to a
unit-diameter sphere in such a way that the poles correspond to and
1 on the complex plane. Then compactify. The arcs passing through
and 1 on the plane are mapped to great circles on the sphere, and
re(z)=1/2 - to the equator. What one gets can be interpreted as the
Bloh sphere of a qubit - the elementary construct in quantum
information theory. Ignore the phase and we are back at ordinary
probability (a projection from entire C-plane to the [0,1] segment, or
any arc anchored at and 1).

best
pg