Relativity and spin 1/2
Howard Wiseman
The Dirac eqn can (and indeed first waS) written as
i (d/dt) psi = H psi
Furthermore, the quantity
P = psi* psi
is the time component of a divergenceless 4-vector. (Here * denotes
Hermitian conjugate).
THese two properties are necessary if psi is to be interpreted as a wave
function for a single particle, not just a field. Supplemented with Dirac's
hole theory, this interpretation seems to be perfectly OK for any particle
of spin 1/2. Second uantization of a fermi field is just a matter of
convenience. (bose fields are not second quantized. They are not particle
wav equations to start with and so they are just quantized)
My question is this: I have read that all known fundamental particles
(fermions for those who like to think of bosons as particles - I don't)
are spin 1/2.
This is stated as a matter of chance. Now, can the wavequations for higher
spin particles be written with the two properties above (e.g. spin 3/2).
Now if they c annot, then surely this is an explanation as to why all
particles are spin 1/2.
I am guessing that the answer to my question is that they cannot, because
it seems intuitive that if you want to form a four vector out of a
bilinear function of a wave function, then that wavefunction has to be
like the square root of a 4 vector in some sense. i.e. it should be like
a tensor of rank 1/2, which is a spinor of a spin 1/2.
Is this a reasonable argument?
Thank you in advance,
Howard Wiseman.
Jason Kodish
|Jason Kodish, -Courage is not
|University of Alberta measured by the
|Dept of Gravitational Engineering success of a man,
| but the odds against
|R -1/2 g R = T (Einstein Field Equation) which he succedes
| un un un
Warren G. Anderson
(Howard Wiseman) writes:
Dirac equation and probability current descriptions deleted
> My question is this: I have read that all known fundamental particles
> (fermions for those who like to think of bosons as particles - I don't)
> are spin 1/2.
> This is stated as a matter of chance. Now, can the wavequations for higher
> spin particles be written with the two properties above (e.g. spin 3/2).
> Now if they c annot, then surely this is an explanation as to why all
> particles are spin 1/2.
I'm not an expert on this, but here is what little I know. Spin 3/2
particles are known as Rarita-Schwinger particles. Because they are
described by a spin 3/2 object (i.e. a vector of spinors) the Lagrange
density would have to be different for such particles and thus they
would satisfy something other than Dirac's equation (just as a scalar
field is given by the Klein-Gordon equation rather than Dirac's equation)
I think. I have been told that it can be shown that any interacting spin
3/2 field has a Green's function which propogates acausally. I believe
work was done on this in the 70's. Nonetheless, I understand that they
necessarily appear in supersymmetric grand unified theories as
supersymmetric partners to the graviton (in supersymmetry, every bosonic
degree of freedom has a corresponding fermionic degree of freedom). The
acausal propogation of these particles seems to have been reconciled in
these theories, but I don't know how.
Hopefully, any mistakes in this information will inspire a real expert
to post corrections. Hope this helps somewhat.
--
########################## _`|'_ ##############################################
## Warren G. Anderson |o o| "... for its truth does not matter, and is ##
## Dept. of Physics ( ^ ) unimaginable." -J. Ashbery, The New Spirit ##
## University of Alberta /\-/\ (ande...@fermi.phys.ualberta.ca) ##
Howard Wiseman
answer.
Perhaps I didn't make my question clear.
First, the two things which I wrote as fundamental for a wave equation were:
1. psi obeys a Schrodinger equation (not necessarily the Dirac eqn).
That is
i(d/dt) psi = H psi
for _some_ Hermitian operator H.
2. that psi* psi be the time cpt of a divergenceless 4 vector.
I suggested that maybe only spin 1/2 particles meet these requirements.
This could explain why all FUNDAMENTAL fermions so far known
(electrons, neutrinos, quarks) are spin 1/2. I know that any particle
with odd spin is a fermion, but that doesn't make it a fundamental particle.
Thanks in advance,
Howard Wiseman.
Ron Maimon
|> Thanks to people who have responded so far, but I'm still looking for an
|> answer.
|> Perhaps I didn't make my question clear.
|>
|> First, the two things which I wrote as fundamental for a wave equation were:
|>
|> 1. psi obeys a Schrodinger equation (not necessarily the Dirac eqn).
|> That is
|> i(d/dt) psi = H psi
|>
|> for _some_ Hermitian operator H.
|>
|> 2. that psi* psi be the time cpt of a divergenceless 4 vector.
|>
|>
|> I suggested that maybe only spin 1/2 particles meet these requirements.
|>
You are right that the only spin for which you can have anthing near to a
convincing wave-equation is for spin 1/2. This is not the reason all known
fundamental fermions are spin 1/2, however.
Your reasoning fails because the requirements you give are not necessary for
a relativistic theory, and are not obeyed by the relativistic theories that
we have today, not even by the modern theory of spin 1/2 particles.
the reason for this is that there is a reinterpretation of the wavefunction
in relativistic quantum mechanics. There is no longer a function psi(x) such
that psi*(x)psi(x) is the probability density of finding a particle at x. There
is no longer an X operator and a P operator which combine to form a Hamiltonian.
Modern relativistic quantum mechanics is very different.
The reason for this is that any theory with interactions in it allows for particle
creation and annihilation. For instance, free electrons just wander around maintaining their momentum. Interacting electrons can, through there interactions
create an electron-positron pair. This process cannot be adequetly represented
by a regular wavefunction.
For example, say I have a single electron moving through an electric field, and suppose that it is described by a wavefunction psi(x). If I now observe that after the electron leaves the electric field it is accompanied by an electron-positron pair, I now need a wavefunction depending on three parameters
psi(x,x',x'') where x is the position of the first electron, x' is the position
of the second electron, x'' is the position of the positron. This is something that cannot happen in nonrelativistic quantum mechanics.
So how do you describe such a system?
well, you need to write down a wavefunction which represents the probability of
having different particle numbers- indeed you need a series of wavefunctions:
psi0, psi1(x,s), psi2( x,s,x',s' ) , psi3(x,s,x',s',x'',s'') , ....
and the interpretation of these wavefunctions are as follows- the number psi0 is
the amplitude that you find that there are no electrons in the system. The wavefunction psi(x,s) is the amplitude that there is one electron at the point x having spin state s ( s takes one four values basically because of the fact that
there are four components to the Dirac equation- or, if you like, because an electron can be in one of four spin states- spin up, spin down, antiparticle spin
up, antiparticle spin down). similarly psi2(x,s,x',s') is the amplitude that
there are two particles present, one is at x and has spin s, the other is at x' and has spin s', and so on.
Now, I should say that people like to write this down in this way (for zero spin):
|psi>= psi0 |0>+ (sum k) psi1(k) a+(k)|k> + (sum k,k') psi2(k,k')a+(k)a+(k')|0> ....
where the symbol |0> denotes the state where there are no particles present, and the operator a+(k) creates a particle with momentum k (people usually work in the momentum representation). So what this says is that the state of the universe is
described by a state |psi> which has within it amplitudes for a whole bunch of configurations. The probability of, for example, finding three particles in the world with momenta 1,4, and 6.6 is |psi3(1,4,6.6)|^2. The probability of finding
no particles in the world is |psi0|^2.
Once you do this, you can write down a relativistically invariant theory for any
spin particles. This is known as second quantization, and is usually presented through a course on field theory.
You might be upset at having to go to all this trouble, when the dirac equation
describes all spin 1/2 particles, and looks just like ordinary nonrelativistic quantum mechanics. The reason all this is necessary is that the Dirac equation
is grossly inadequate as a wave-equation. The hamiltonian is not positive definite, and the charge is. This means that (if you work it out) the particles described by the dirac equation all have the same charge, and are either of positive or negative energy. Dirac got out of this difficulty with his famous electron sea, but this requires a theory with more than one particle anyway. Indeed, IMO, this was Dirac's great achievement with the Dirac equation- recognizing the need to make it into a many-particle
e
quation.
The problem with having negative energies is that when you introduce an interaction with, for instance, the electromagnetic field, the electron will spiral down into lower and lower energy states, a very unstable state indeed!
|> This could explain why all FUNDAMENTAL fermions so far known
|> (electrons, neutrinos, quarks) are spin 1/2. I know that any particle
|> with odd spin is a fermion, but that doesn't make it a fundamental particle.
|>
|> Thanks in advance,
|> Howard Wiseman.
|>
It is certainly not true that every fundamental particle is spin 1/2. The photon
has spin 1, and so do the weak bosons, and the gluons. The graviton (very likely) has spin 2.
the reason all the theories we know today have spin 1/2 particles interacting with spin 1 particles is that we require that the theories be renormalizable. This is a very strict condition.
In order to make sense of a field theory, you must consider it on a lattice (or do something similar in momentum space) and then take the lattice spacing down to zero. As you take the lattice spacing to zero, you must adjust the constants in the theory so that it yeilds finite amplitudes for certain transitions whose rate you know from experiment. You then take the limit in such a way that you get the right answer for those processes, and you find that by taking this limit you get the right answer for any
p
rocess. In order for this to happen the theory must be renormalizable.
The only known nontrivial renormalizable theories are self-interactions (of particles with other particles of the same type) and certain types of interactions between spin 1/2 and spin 1 particles. The theories based on these interactions are known as gauge theories, and they successfully explain the weak strong and electromagnetic forces, but not gravitation.
So ultimately the reason that the particles we can see have spin 1/2 is that these are the only theories that make sense, but this has nothing (directly) to do with the fact that the dirac equation can be used as a (bad) wave equation.
Ron MaimonUnread news in sci.physics 134 articles + 1040 old
Lawrence R. Mead
: Thanks to people who have responded so far, but I'm still looking for an
What do you mean by *fundamental* ? Why do you think that a photon (for
example) which is a Boson is less fundamental?
--
Lawrence R. Mead (lrm...@whale.st.usm.edu) | ESCHEW OBFUSCATION !
Associate Professor of Physics
Howard Wiseman
I do not think that a photon is a less fundamental entity than an electron.
However, I would not call the photon a particlein the same sense.
I know that this is a matter of semantics, but I deliberately avoided this
issue by wording my question as "why all fundamental FERMIONS" are spin 1/2.
The point I am trying to elucidate is: as far as I can see, there is no reason
why Dirac's equation cannot be given his original interpretation, supplemented
by his later idea of holes and electron seas.
Now, if the Dirac eqn is the only true relativistic wave eqn (satisfying
conditions 1 and 2 above) then this would explain why all fundamental fermions
are spin 1/2: because that is the only possible spin for a fundamental particle.
Fundamental bosons do not have wavefunctions. They arise from the quantisation
of classical fields. Fermion fields cannot be derived this way. The anticommutator
cannot come from canonical quantisation. Fermi fields are a mathematical; way to
represent many particle dynamics (including the electron sea etc). This is why
fundamental bosons can have any integral spin (they are not particles with
wavefunctions), but all fundamental fermions are spin 1/2.
If anybody could point out a fundamental fermion that is not spin 1/2,
or tell me where to find a wave equation satisfying 1 and 2 for a fermion of spin 3/2
or 5/2 or whatever,
or tell me if there is any problem with sticing to Dirac's interpretation,
or tell me where there is a proof that only spin 1/2 particles can have valid
relativistic wavefunction equations (satisfying 1 and 2),
or find any other fault in or support for my argument, I would be most grateful.
Howard Wiseman, U.Q. Australia.
: --
Lawrence R. Mead
: Lawrence R. Mead (lrm...@whale.st.usm.edu) wrote:
: : Howard Wiseman (wis...@kelvin.physics.uq.oz.au) wrote:
: : : I suggested that maybe only spin 1/2 particles meet these requirements.
: : : This could explain why all FUNDAMENTAL fermions so far known
: : : (electrons, neutrinos, quarks) are spin 1/2. I know that any particle
: : : with odd spin is a fermion, but that doesn't make it a fundamental particle.
: : : Thanks in advance,
: : : Howard Wiseman.
: : What do you mean by *fundamental* ? Why do you think that a photon (for
: : example) which is a Boson is less fundamental?
: I do not think that a photon is a less fundamental entity than an electron.
: However, I would not call the photon a particlein the same sense.
Why not? It has the same particle-like properties as electrons do when the
proper experiments are done (they carry momentum, etc.)
: I know that this is a matter of semantics, but I deliberately avoided this
: issue by wording my question as "why all fundamental FERMIONS" are spin 1/2.
AS long as we are talking semantics, what do you mean by "fundamental"?
: The point I am trying to elucidate is: as far as I can see, there is no reason
: why Dirac's equation cannot be given his original interpretation, supplemented
: by his later idea of holes and electron seas.
Because, as is well-known, you must then accept particles which move backward
in time ( the negative energy ones) , and there cannot be a "Dirac" sea for
Bosons. The only consistent spin 1/2 theory is a many-body theory (ie., a
quantum field).
: Now, if the Dirac eqn is the only true relativistic wave eqn (satisfying
: conditions 1 and 2 above) then this would explain why all fundamental fermions
: are spin 1/2: because that is the only possible spin for a fundamental particle.
It is not. Spin 3/2 particles are called Rarita-Schwinger particles and are
(to my mind) no less fundamental.
: Fundamental bosons do not have wavefunctions. They arise from the quantisation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Who said? All quantum-mechanical particles or systems of them have
wavefunctions (ie., probability amplitudes) associated with them. The state-
vector is *the* fundamental idea of QM.
: of classical fields. Fermion fields cannot be derived this way. The anticommutator
: cannot come from canonical quantisation. Fermi fields are a mathematical; way to ^^^^^^^^^^^^^^^^^^^^^^
Neither do *commutation* relations. Both types of field commutations are
postulated in canonical quantization procedure : a choice of one or the
other must be made . Both ,however are fully consistent with the Schwinger
action principle. Thus, neither field type "comes from" canonical procedure.
: represent many particle dynamics (including the electron sea etc). This is why
: fundamental bosons can have any integral spin (they are not particles with
: wavefunctions), but all fundamental fermions are spin 1/2.
: If anybody could point out a fundamental fermion that is not spin 1/2,
: or tell me where to find a wave equation satisfying 1 and 2 for a fermion of spin 3/2 (see, for example, Intro. to Quantum Field theory by Roman for the
RArita-Schwinger equation or the Proca equation for spin 1 massive Bosons)
: or 5/2 or whatever,
Ron Maimon
|>
|> The point I am trying to elucidate is: as far as I can see, there is no reason
|> why Dirac's equation cannot be given his original interpretation, supplemented
|> by his later idea of holes and electron seas.
|> Now, if the Dirac eqn is the only true relativistic wave eqn (satisfying
|> conditions 1 and 2 above) then this would explain why all fundamental fermions
|> are spin 1/2: because that is the only possible spin for a fundamental particle.
|>
|> Fundamental bosons do not have wavefunctions. They arise from the quantisation
|> of classical fields. Fermion fields cannot be derived this way. The anticommutator
|> cannot come from canonical quantisation. Fermi fields are a mathematical; way to
|> represent many particle dynamics (including the electron sea etc). This is why
|> fundamental bosons can have any integral spin (they are not particles with
|> wavefunctions), but all fundamental fermions are spin 1/2.
There is nothing sacred about commutator quantization! you are assuming that
because the equations of field quantization involve the commutator of the field
and its canonically conjugate momentum that this relationship is of the same
sort as the relationship
[X,P] = i
this is not true, otherwise anticommutator quantization would indeed be
fundamentally different from any other type of quantum mechanics.
The reason for the relationship [x,p]=i is because the momentum in quantum
mechanics is the generator of displacements. What this means is that if I have
a system with a wavefunction |psi> and I look at the same system moved a small
amount e to the left, that system is described by the state
|psi> - i e P |psi>
that leads immediately to the realization that the eigenstates of x must be transformed by p in the following way:
- i e P |x> = |x+e> - |x>
and this becomes a more accurate statement as e --> 0. this can be used to derive
the commutation relation between x and p. Just notice that
- i X e P |x> + i e P X |x> = X ( |x+e> - |x> ) - x ( |x+e> - |x> ) = e |x>
dividing by e, and letting e tend toward zero so that the manipulations are
legitemate, we end up with
-i ( X P - P X ) |x> = |x>
and since this operator is the identity when operating on the complete set of
states |x>, it must be the identity in any basis, and so
X P - P X = i
this derivation is flawless in nonrelativistic quantum mechanics, but there is no
analogous derivation of the relations for fields: [ field, conjugate ] = i deltafn
and this is good, otherwise you would be right about anticommutator quantization
being illegitemate.
the reason there is no such derivation is that the `conjugate momenta' don't
necessarily generate field motions like the momenta generate x motions. Well, they
do generate field motions in the boson case- since they obey the right commutation
relations- but thats sort of a coincidence, since it doesn't happen in the fermion
case.
|> If anybody could point out a fundamental fermion that is not spin 1/2,
|> or tell me where to find a wave equation satisfying 1 and 2 for a fermion of spin 3/2
|> or 5/2 or whatever,
|> or tell me if there is any problem with sticing to Dirac's interpretation,
|> or tell me where there is a proof that only spin 1/2 particles can have valid
|> relativistic wavefunction equations (satisfying 1 and 2),
|> or find any other fault in or support for my argument, I would be most grateful.
|>
|> Howard Wiseman, U.Q. Australia.
There is a huge problem with Dirac's interpretation- it isn't a complete
description of nature! For example, I have no idea how to calculate the amplitude
for particle antiparticle creation in Dirac's theory, since you need to start off
with a one particle state and end up in a three particle state, and I don't see
that happenning with his formalism.
You might say that the initial state is not a one particle state, but a many particle state, since the vacuum is full of negative energy positive charge
particles. This doesn't make the problem any more tractable however. I am not
so sure it is impossible to make this work, what I am sure of is, that once you
do make it work, that it will be equivalent to the standard second quantized field
theory, if it can be made to work in the first place.
There are perfectly good free field equations for spin 3/2. I will post them
sometime tomorrow, cause I have to derive them, but theyre not hard to get.
Remember, the only difference between bosons and fermions is in their commutation
relations, and that isn't as big a difference as you make it out to be
Ron Maimon
Howard Wiseman
: Howard Wiseman (wis...@kelvin.physics.uq.oz.au) wrote:
: : : Howard Wiseman (wis...@kelvin.physics.uq.oz.au) wrote:
: : : : I suggested that maybe only spin 1/2 particles meet these requirements.
: : : : This could explain why all FUNDAMENTAL fermions so far known
: : : : (electrons, neutrinos, quarks) are spin 1/2. I know that any particle
: : : : with odd spin is a fermion, but that doesn't make it a fundamental particle.
: : : : Thanks in advance,
: : : : Howard Wiseman.
: : : What do you mean by *fundamental* ? Why do you think that a photon (for
: : : example) which is a Boson is less fundamental?
: : I do not think that a photon is a less fundamental entity than an electron.
: : However, I would not call the photon a particlein the same sense.
: Why not? It has the same particle-like properties as electrons do when the
: proper experiments are done (they carry momentum, etc.)
: : I know that this is a matter of semantics, but I deliberately avoided this
: : issue by wording my question as "why all fundamental FERMIONS" are spin 1/2.
: AS long as we are talking semantics, what do you mean by "fundamental"?
unable to be broken down. quarks are fundamental (as far as we know), so are
electrons. If you think this is a matter of semantics, tell me a spin 3/2
particle (by name) which you would justify calling fundamental in some sense.
: : The point I am trying to elucidate is: as far as I can see, there is no reason
: : why Dirac's equation cannot be given his original interpretation, supplemented
: : by his later idea of holes and electron seas.
: Because, as is well-known, you must then accept particles which move backward
: in time ( the negative energy ones) , and there cannot be a "Dirac" sea for
: Bosons. The only consistent spin 1/2 theory is a many-body theory (ie., a
: quantum field).
Is there any reason I can't accept particles going backwards in time? The Dirac
sea makes the predictions of his theory exactly the same as that of field theory
with particles and antiparticles.
Of course Bosons do not have a Dirac sea. Fundamental bosons are not particles
with wavefunctions, so the concept cannot even arise. I know it won't work becuase
they would condense into the -ve energy states. The "antiparticles" for bosons
arise from the canonical quantization of the bose field which is a classical
field NOT A WAVEFUNCTION.
: : Now, if the Dirac eqn is the only true relativistic wave eqn (satisfying
: : conditions 1 and 2 above) then this would explain why all fundamental fermions
: : are spin 1/2: because that is the only possible spin for a fundamental particle.
: It is not. Spin 3/2 particles are called Rarita-Schwinger particles and are
: (to my mind) no less fundamental.
: : Fundamental bosons do not have wavefunctions. They arise from the quantisation
: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: Who said? All quantum-mechanical particles or systems of them have
: wavefunctions (ie., probability amplitudes) associated with them. The state-
: vector is *the* fundamental idea of QM.
Yes, you are perfectly correct. However, the thing which is quantised to give
photons for example, is not a wavefunction. It is a classical field (the EM field
in this case). It is not quantum mechanical at all. Once you have quantized
it then you have state vectors in Hilbert space (which is the fock space of the
photon fiedl). If you want to call these state vectors wavefunctions, you may,
but the underlyuing classical field is not a wavefunction.
: : of classical fields. Fermion fields cannot be derived this way. The anticommutator
: : cannot come from canonical quantisation. Fermi fields are a mathematical; way to ^^^^^
^^^^^^^^^^^^^^^^^
: Neither do *commutation* relations. Both types of field commutations are
: postulated in canonical quantization procedure : a choice of one or the
: other must be made . Both ,however are fully consistent with the Schwinger
: action principle. Thus, neither field type "comes from" canonical procedure.
Here I am not an expert. I know that either choice can be consistently carried through.
However, if you chose anticommutation relations for your field when you quantize
it, in order to generate the evolution of the field operator. you still use the
canonical commutation relations. That is you still use a commutator in Heisenberg's
equation of motion
(d/dt) a = i[H,a]
Now it seems strange to me that you should chose anticommutation relations to replace
the classical Poisson bracket in some situations, and commutation relations
elsewhere. It is always necessary to use the commutation relations at the point of
the Heisenberg's equations, so it makes sense to use it always. That is,
CANONICAL QUANTIZATION of a field always gives you bosons.
You can still have fields obeying anticommutation relations as long as you realize
that they arise not from canonicl quantization of a classical field, but rather
from a convenient representation of a system of interacting paritcles which obey
Fermi statistics. (Obeying Fermi statistics is necessary to enable a Dirac sea).
: : represent many particle dynamics (including the electron sea etc). This is why
: : fundamental bosons can have any integral spin (they are not particles with
: : wavefunctions), but all fundamental fermions are spin 1/2.
: : If anybody could point out a fundamental fermion that is not spin 1/2,
: : or tell me where to find a wave equation satisfying 1 and 2 for a fermion of spin 3/2
: (see, for example, Intro. to Quantum Field theory by Roman for the
: RArita-Schwinger equation or the Proca equation for spin 1 massive Bosons)
I know of the existence of the RS equation and the Proca equation.
However, as far as I can make out, these are not valid wavefunction equations,
According to the two criteria I began with. Specifically, the quantity
psi* psi
is not the time component of a conserved four vector. This is the essential point I was making.
t is certainly not the case for the Proca equation. For the RS equation, the quantity
psi^{mu}* psi_{mu}
may be that timne component. However, the contraction (using the SR metric tensor) is not
part of QM. If each element of the wavefunction is to be interpreted as the amplitude for the
particle to be in that state, the the total probability density should be
psi^{mu}* psi^{mu} (with summation)
which is different. That is, the sum of the modulus squared of each amplitude.
Please correct me if I'm wrong, but the essential point as I see it is that the correct prob.
density can only be constructed from psi* psi if psi is a spinor of rank 1. That is to say,
only if psi is the wavefunction for a particle of spin 1/2.
: : or 5/2 or whatever,
: : or tell me if there is any problem with sticKing to Dirac's interpretation,
: : or tell me where there is a proof that only spin 1/2 particles can have valid
: : relativistic wavefunction equations (satisfying 1 and 2),
: : or find any other fault in or support for my argument, I would be most grateful.
: : Howard Wiseman, U.Q. Australia.
: : : --
: : : Lawrence R. Mead (lrm...@whale.st.usm.edu) | ESCHEW OBFUSCATION !
: : : Associate Professor of Physics
Howard Wiseman again. above requests still hold.
Lawrence R. Mead
: Lawrence R. Mead (lrm...@whale.st.usm.edu) wrote:
: : Howard Wiseman (wis...@kelvin.physics.uq.oz.au) wrote:
: : : : Howard Wiseman (wis...@kelvin.physics.uq.oz.au) wrote:
: : : : : This could explain why all FUNDAMENTAL fermions so far known
: : : : : (electrons, neutrinos, quarks) are spin 1/2. I know that any particle
: : : : : with odd spin is a fermion, but that doesn't make it a fundamental particle.
: : : : : Thanks in advance,
: : : : : Howard Wiseman.
: : : issue by wording my question as "why all fundamental FERMIONS" are spin 1/2.
: : AS long as we are talking semantics, what do you mean by "fundamental"?
: unable to be broken down. quarks are fundamental (as far as we know), so are
I presume that (unable to be broken down ) means particles that can decay?
: Is there any reason I can't accept particles going backwards in time? The Dirac
Because it leads to logic paradoxes.
: sea makes the predictions of his theory exactly the same as that of field theory
: with particles and antiparticles.
No, it does not. For example, pair production and anihilation as well as
many others.
: arise from the canonical quantization of the bose field which is a classical
: field NOT A WAVEFUNCTION.
It is too a wavefunction: the E and B (or A and V) satisfy the vector
wave equation. In what sense is the quantized fields NOT a wavefunction?
: photons for example, is not a wavefunction. It is a classical field (the EM field
: in this case). It is not quantum mechanical at all. Once you have quantized
: it then you have state vectors in Hilbert space (which is the fock space of the
: photon fiedl). If you want to call these state vectors wavefunctions, you may,
: but the underlyuing classical field is not a wavefunction.
See above.
: : other must be made . Both ,however are fully consistent with the Schwinger
: : action principle. Thus, neither field type "comes from" canonical procedure.
: Here I am not an expert. I know that either choice can be consistently carried through.
: However, if you chose anticommutation relations for your field when you quantize
: it, in order to generate the evolution of the field operator. you still use the
: canonical commutation relations. That is you still use a commutator in Heisenberg's
: equation of motion
: (d/dt) a = i[H,a]
: Now it seems strange to me that you should chose anticommutation relations to replace
It is no stranger to use { , } instead. Both lead to perfectly consistent
field equations, etc. Consult any field theory text.
: the classical Poisson bracket in some situations, and commutation relations
: elsewhere. It is always necessary to use the commutation relations at the point of
: the Heisenberg's equations, so it makes sense to use it always. That is,
: CANONICAL QUANTIZATION of a field always gives you bosons.
: Howard Wiseman again. above requests still hold.
--many remarks edited out-- I don't think you will get much agreement with
trying to preserve a formulation of relativistic QM that does not
cover all of the phenomena we see.
Ralf Muschall
>Lawrence R. Mead (lrm...@whale.st.usm.edu) wrote:
>: Howard Wiseman (wis...@kelvin.physics.uq.oz.au) wrote:
>: : Thanks to people who have responded so far, but I'm still looking for an
>: : answer.
>: : Perhaps I didn't make my question clear.
>
>: : First, the two things which I wrote as fundamental for a wave equation were:
>
>: : 1. psi obeys a Schrodinger equation (not necessarily the Dirac eqn).
>: : That is
>: : i(d/dt) psi = H psi
Just extract all time derivatives, call the arguments which are differenciated
wrt. time `components of psi' and put them to the left-hand side of the eqn.
I would prefer a Dirac-like equation, i.e.
\nabla^{AA'}\Psi_{ABC...} = 0
which has the advantage to be manifestly covariant.
>
>: : for _some_ Hermitian operator H.
>
> : : 2. that psi* psi be the time cpt of a divergenceless 4 vector.
This depends on how you define the adjoint operation and the multiplication
for multi-component quantities.
I agree that one could wish that there should exist a real divergenceless 4-vector
which depends linearly on the tensor product of Psi and adj(Psi), where
adj is _any_ antilinear operation.
But this is only a charge-flow vector. A divergenceless energy-momentum tensor
seems to be more important.
[lines deleted]
>
>Fundamental bosons do not have wavefunctions. They arise from the quantisation
>of classical fields.
You mean that we didn't use wavefunctions before the quantization was done.
They have the same right to possess wavefunctions as any other particle.
>Fermion fields cannot be derived this way. The anticommutator
>cannot come from canonical quantisation. Fermi fields are a mathematical; way to
>represent many particle dynamics (including the electron sea etc). This is why
>fundamental bosons can have any integral spin (they are not particles with
do they exist (beyond Maxwell and Weyl)?
>wavefunctions), but all fundamental fermions are spin 1/2.
>
>If anybody could point out a fundamental fermion that is not spin 1/2,
>or tell me where to find a wave equation satisfying 1 and 2 for a fermion of spin 3/2
>or 5/2 or whatever,
>or tell me if there is any problem with sticing to Dirac's interpretation,
>or tell me where there is a proof that only spin 1/2 particles can have valid
>relativistic wavefunction equations (satisfying 1 and 2),
>or find any other fault in or support for my argument, I would be most grateful.
>
>Howard Wiseman, U.Q. Australia.
>
>
There are other, more serious problems with equations for higher spin fields:
The equation I wrote above (it is the Dirac eqn. for the neutrino field,
the Maxwell eqn. for the spin=1 and the Bianchi identity for gravity)
has rather restricive integrability conditions for greater spins.
E.g. for spin=2, the only solution in the general case is the gravity field itself.
(See Penrose/Rindler, Spinors & Spacetime, Vol. 1, chapter 5, section 8
and citations there).
Ralf
--
--
#include <disclaimer.h>
PGP 2.1 key obtainable with finger p...@hpux.rz.uni-jena.de
(or from public key servers)
john baez
>There are other, more serious problems with equations for higher spin fields:
>The equation I wrote above (it is the Dirac eqn. for the neutrino field,
>the Maxwell eqn. for the spin=1 and the Bianchi identity for gravity)
>has rather restricive integrability conditions for greater spins.
>E.g. for spin=2, the only solution in the general case is the gravity field itself.
>(See Penrose/Rindler, Spinors & Spacetime, Vol. 1, chapter 5, section 8
>and citations there).
I'm not sure what you mean here and I don't have Penrose and Rindler to
hand. But what I do know is that there is a perfectly fine linear wave
equation describing particles of any spin j one likes. (See Wigner's
classification of unitary representations of the Poincare group - Ann.
Math. 40 (1939), p. 149.) The troubles start when one wishes to include
nonlinear interactions. General relativity can be thought of as the
equations describing a self-interacting massless spin-2 particle,
although I'd prefer not to think of things this way, since it's an
essentially perturbative viewpoint.
to...@cc.usu.edu
You are both essentially right.
There are perfectly good field equations for free fields
of any spin in Minkowski space. However, these equations are not compatible
with curved spacetime for spin >2 in the sense that they imply (as
integrability-type conditions) there are algebraic relations between the
field and pieces of the curvature that severely restrict the solution space of
the field equations and/or the spacetime. Note that the field equations are
still linear, they simply do not generalize nicely to curved backgrounds. The
integrability conditions are sometimes called "Buchdahl conditions".
Howard Wiseman
: [X,P] = i
This is my point. It is not a coincidence. For bosons, the conjugate field momentum
does what it is supposed to do becasue the quantization is achieved by the usual
(canonical quantization) method. The "second quantization" of fermions is not
a quantization at all. It is merely a way of doing bookkeeping. In that case, of
course the conjugate field is not really a conjugate field, because you are not
really doing quantization.
: |> If anybody could point out a fundamental fermion that is not spin 1/2,
: |> or tell me where to find a wave equation satisfying 1 and 2 for a fermion of spin 3/2
: |> or 5/2 or whatever,
: |> or tell me if there is any problem with sticing to Dirac's interpretation,
: |> or tell me where there is a proof that only spin 1/2 particles can have valid
: |> relativistic wavefunction equations (satisfying 1 and 2),
: |> or find any other fault in or support for my argument, I would be most grateful.
: |>
: |> Howard Wiseman, U.Q. Australia.
: There is a huge problem with Dirac's interpretation- it isn't a complete
: description of nature! For example, I have no idea how to calculate the amplitude
: for particle antiparticle creation in Dirac's theory, since you need to start off
: with a one particle state and end up in a three particle state, and I don't see
: that happenning with his formalism.
: You might say that the initial state is not a one particle state, but a many particle state, since the vacuum is full of negative energy positive charge
: particles. This doesn't make the problem any more tractable however. I am not
: so sure it is impossible to make this work, what I am sure of is, that once you
: do make it work, that it will be equivalent to the standard second quantized field
: theory, if it can be made to work in the first place.
I am not disputing that the anticommutator treatment of fermions is a perfectly correct
way, and probably the only practical way, to treat the problem. All I am saying is,
as you seem to admit yourself, that it is a derived concept from Dirac's negative
energy sea. That is, it is dirac's idea made workable. Remeber, Dirac predicted
antiparticles and annihilation/ creation of pairs using his theory, not a
"second quantized" electron field.
: There are perfectly good free field equations for spin 3/2. I will post them
: sometime tomorrow, cause I have to derive them, but theyre not hard to get.
I look forward to seeing your proof that they are perfectly correct.
My problem with all representations that I've seen so far is that
psi* psi is not the time cpt of a conserved 4 vector.
For example, in the Rarita-Schwinger representation, psi is a 4-vector,
each component of which is a Dirac bispinor. Now if you take the inner
product of this vecot with its adjopint, you maybe do get a quantity
with the right properties. However, that process (inner product) is not
part of QM. The scalar product is simply defined by the product of each
component with its complex conjugate. This product is not the same as
the vector poduct, because g00 = -1.
: Remember, the only difference between bosons and fermions is in their commutation
: relations, and that isn't as big a difference as you make it out to be
: Ron MaimoN
Ron Maimon
|>
|> This is my point. It is not a coincidence. For bosons, the conjugate field momentum
|> does what it is supposed to do becasue the quantization is achieved by the usual
|> (canonical quantization) method. The "second quantization" of fermions is not
|> a quantization at all. It is merely a way of doing bookkeeping. In that case, of
|> course the conjugate field is not really a conjugate field, because you are not
|> really doing quantization.
|>
thats not a good way of thinking about it.
you have no idea what a conjugate field is "supposed" to do, or what commutation
relations its "supposed" to have, because there is absolutely no necessary
relationship between the field and the conjugate momentum.
the kinematic momentum is _defined_ as the generator of translations, the
conjugate momentum is just _named_ that, its not really a momentum, it doesn't
generate translations. It's just a coincidence that the boson commutator of the
field and the field momentum looks like the commutator between the one particle
position and the one particle momentum.
the real reason why the field and the conjugate momentum have nice looking
commutators in the boson case is that the heisenburg equation of motion
d/dt phi = i[H,phi]
must be consistent with the equation of motion
(d/dt)^2 phi - (grad)^2 phi = m^2 phi
with the hamiltonian being given by the form
H = (integral) 1/2 (dphi/dt)^2 + 1/2 (grad phi)^2 + 1/2 m^2 phi^2
this can be done in many ways, corresponding to the many, many methods of doing
statistics for free particles.
the two _simplest_ ways are to assume
[phi(x) , d/dt phi(x') ] = delta ( x-x')
or
{phi(x) , d/dt phi(x') } = delta ( x-x')
these correspond to bose einstein and fermi-dirac statistics respectively.
Since phi is an observable, the first option is the only one consistent with
causalty, but both options are _valid quantization procedures_ just because one
looks more like ordinary one particle quantum mechanics in that it involves
commutators and not anti commutators doesn't make it "correct". you just are
stuck thinking that the statement
[p,q]=i
is a "fundamental axiom" of quantum mechanics or some such thing, when actually
its an obvious relationship that could be derived from general principles (as I
tried to show last post).
Ron Maimon
Phil Gibbs
Supergravity is an example of a relativistic theory with spin 3/2 particles.
They are known as "gravitinos".
Howard Wiseman
: Supergravity is an example of a relativistic theory with spin 3/2 particles.
: They are known as "gravitinos".
I meant an example of a particle which has actually been observed, or at least implied
by other observations. As far as I knew, a graviton is still a hypothetical particle,
let alone its supposed supersymeetric counterpart.
Howard Wiseman
: or
: [p,q]=i
: Ron Maimon
I can see your point of view. However I don't think that that means that my point
of view is incorrect. Rather than debating about what should be, I would be greatful
if you could answer some factual questions which are not open to interpretation
1. Is it not true that the quantized field description of spin 1/2 fermions is
equivalent to Dirac's hole theory. (Although the former is probably a much more
practical way to deal with annihilation/creation etc.)
2. Can you write the Rarita-Schwinger (or any other form) of the wave equation
for a spin 3/2 particle for example, in the Schrodinger form?
That means, as I have explained before,
(a) (d/dt) psi = iH psi
(b) psi* psi is the time cpt of a divergenceless 4 vector.
Now, psi* here means transpose plus complex conjugation. That is to say,
psi* psi equals the sum of the modulus squared of each component of psi.
This is not the same as
g_{uv} psi*^u psi^v
which is what seems to me to arise from the RS equation for spin 3/2 particles.
On the basis of the answer of these two questions, I am of course forming my
own biassed (and probably outdated) opinion about the nature of quantum
field theory. You (and everyone else who has replied) may disagree about my
interpretation, and I am happy to listen to the reasons. However, I would
still like a straight answer to the two questions I have above, regardless
of whether you think a straight answer would mislead me or not. I am not an
expert in QFT, which is why I am asking these questions. Would someone who
is an expert please answer my questions with an open mind.
Thank you.
Ron Maimon
|>
|> I can see your point of view. However I don't think that that means that my point
|> of view is incorrect. Rather than debating about what should be, I would be greatful
|> if you could answer some factual questions which are not open to interpretation
|>
These are, of course, the only questions worth thinking about in physics!
|> 1. Is it not true that the quantized field description of spin 1/2 fermions is
|> equivalent to Dirac's hole theory. (Although the former is probably a much more
|> practical way to deal with annihilation/creation etc.)
|>
I would say no.
the reason for this is that I don't know any complete statement of the dirac
theory.
example. If I scatter an electron with a momentum k off another electron with
a momentum -k (center of mass system) there is an amplitude for creating a state
with four electrons present- two electrons and two positrons. What is this
amplitude, and what is its functional dependence on the momenta of the outgoing
particles?
this is a perfectly legitemate experimentally verifiable (and verified) process,
and I don't know how you would calculate the probability that it will happen
in Dirac's hole theory. I admit that you can sort-of qualitatively see that it
will happen, but thats not a probability.
I admit that I have always thought of Dirac's hole theory as a qualitative
explanation of particle pair production that made the whole notion palatable
to the early quantum physicists. I never saw it as a complete calculational
scheme that allows you to figure out the amplitude of any process. I strongly
suspect that there is no way to make it into a quantitative scheme capable of
answering the question I posed above without second quantizing it.
and using fermi-dirac quantization.
|> 2. Can you write the Rarita-Schwinger (or any other form) of the wave equation
|> for a spin 3/2 particle for example, in the Schrodinger form?
|>
|> That means, as I have explained before,
|>
|> (a) (d/dt) psi = iH psi
|>
|> (b) psi* psi is the time cpt of a divergenceless 4 vector.
|>
|> Now, psi* here means transpose plus complex conjugation. That is to say,
|> psi* psi equals the sum of the modulus squared of each component of psi.
|> This is not the same as
|>
|> g_{uv} psi*^u psi^v
|>
|> which is what seems to me to arise from the RS equation for spin 3/2 particles.
|>
I emphasize that the "schroedinger form" for equations of motion does not exist
in a theory of interacting particles.
However, it is possible to write down a "Schroedinger equation" for particles of
any spin, as long as they are completely free from interactions (and therefore
unobservable).
I will do it for a spin zero particle, then for a spin 1/2 particle, and by then
it should be clear what to do for any spin. I remind you, however, that these
equations are in their very essence equations that describe free particles, and
there is no way of introducing interactions into the theory without modifying it
into a second quantized form or destroying relativistic invariance.
spin 0
------
If I have a single free spin zero particle, it is completely described by its
momemtum k. Let me call the states in which it has a definite momentum |k>. The
fact that this is a complete description is a statement that any state can be
written as a superposition of these states
|psi> = (integral) f(k) |k>
It also means that if I know the time evolution of these states I know the time
evolution of any state.
so a complete "equation of motion" or "schroedinger equation" or whatever people
call it, is simply a description of the time evolution of the most general state,
or in this case, of the states |k>
Fortunatly, there is only one covariant way to specify how these states evolve in
time. It involves the observation that K is conserved!! therefore the states |k>
must evolve into themselves after a time t, or into phaze multiples of themselves.
This means that the states |k> are eigenstates of the hamiltonian.
H |k> = E(k) |k>
(this is all pretty obvious for a free particle)
and the form of E(k) completely determines the evolution of the system.
define E(0)= m.
now if I want to find E(k), all I have to do is perform a lorentz transformation
which takes the state |0> with zero momentum into the state |k>, and see what
happens to the energy. The calculation gives the result that
E(k) = sqrt( m^2 + k^2)
and this means that the hamiltonian is uniquely determined for a relativistic
theory of one free spin zero particle.
The corresponding schroedinger equation in position space is
i d/dt psi(x) = sqrt( m^2 - (grad)^2 ) psi
and this is a _perfectly good_ equation _but_only_for_a_free_particle_. You might
object to the appearance of a sqrt of a derivative ( What in the world could that
mean?) but it is perfectly acceptable- this equation is just another way of
writing the energy momentum relationship for a free particle, and I showed you
above how to solve the initial value problem for it.
Given psi(x,0), fourier transform it into psi(k,0), then you can solve the
equation
id/dt psi(k,t) = E(k) psi(k,t)
the solution is just
psi(k,t) = exp (-iE(k)t) psi(k,0)
and now just fourier transform back into position space to find psi(x,t)
still this equation looks funny, it doesn't look like other equations in physics.
It has a very strange operator in it acting as a hamiltonian. What gives?
The reason this equation looks funny is becuase it is funny. It is nonlocal,
meaning that a disturbance in this equation travels faster than light. This is
no problem so long as there are no interactions, remember, because then the
particle can't be used to send signals, but this shows that it is impossible
to add in any sort of interaction, since then we could detect the particle, and
use it to send messages back in time and all that nonsense.
but it is still the equation obeyed by a spinless meson when it is far away from
anything else- it is the correct wave equation for a spin zero particle. It just
does not describe the meson when it is interacting with anything else. It is also
the equation obeyed by a single solitary (hypothetical) Higgs boson when it is
far from anything else.
notice that psi*(x)psi(x) is conserved since the function E(p) is real (or if you like, since the hamiltonian is hermitian).
Spin 1/2
--------
the way to do this is analogous. You label each of the states with definite
momentum and spin by there momentum and their spin.
|k,s>
k is a continuous label, s is either 1 or 2 corresponding to the two linearly
independent spin states. The hamiltonian conserves both momentum and angular
momentum- so the correct energy function is
E(k,s)= sqrt( m^2 + k^2)
and the schroedinger equation can be written
i d/dt psi(k,s) = sqrt( m^2 + (grad)^2 ) psi(k,s)
again, this can only work for a free electron.
What does this have to do with the Dirac Equation?
the solutions of the dirac equation can be written as
/ psi1(x) \
| psi2(x) |
psi(x) = | psi3(x) |
\ psi4(x) /
like Dirac did. They can be fourier decomposed to find psi(k) in terms of the
four dirac components. Then we find that of the four linearly independent states
at any k, two have positive energy and two have negative energy. If we write the
states of positive energy at a given k P1(k) and P2(k) and write the states of
negative energy as N1(k) and N2(k), we can write any state as
psi(k,1) P1(k) + psi2(k,2) P2(k)
there are no amplitude for an electron to be in a negative energy state- those
are all occupied.
If I make a good choice of P1 and P2 these are eigenstates of the spin in the z
direction. and then the psi in the above are the same as the psi(k,s) as in the
"schroedinger equation" I wrote out before.
So you see the Dirac equation is nothing special if you interpret it as a wave-
equation. Its just what you would get if you tried to write down an equation for
spin 1/2 free particles. It doesn't solve the problem of faster-than-light travel
time, so again, you can't add interactions consistently.
I should say that normally you don't do what I did and throw away the two states
N1 and N2, but thats the only way I can make any sense of Dirac's prescription
of saying the negative energy states are filled.
I also note that it is possible to make sure the travel time is less than that of
light (and also to make sense therefore of the interactions) if you allow the
negative energy states to exist. But this, of course, leads to awful things.
Electrons will spontaniously enter the negative energy states and radiate all this
energy away and so on.
The way Dirac used this is really neat- he ignored the problems with solving the
time dependent equation- the fact the electrons would occupy the negative energy
states. Instead, he solved the time-independent equation for all the energy
eigenstates a single particle would have. He only threw away the negative energy
states after he got the spectrum.
For example, the hydrogen ground state in the dirac theory is built both from
positive and negative frequency fourier components, but this doesn't mean its
energy is negative.
This gives results in good agreement with experiment but its _extremely_ ad-hoc.
You need a good time-dependent theory.
spin 3/2
--------
I don't need to go through this do I?
again, you can write down a perfectly good schroedinger equation, but it will not
work with interactions.
Maybe there is a phenomenological way to work out the energy levels of hydrogen
if the electron had spin 3/2 much like Dirac worked out the energy levels of
hydrogen phenomenologically using his equation. I suspect you need to do nothing
more than solve the spin 3/2 field equation (the equation that corresponds to
the dirac equation) in a time-independent setting, and then throw away the
negative energy states.
I hope this was a satisfying enough critique of the Schroedinger equation in the
context of relativistic mechanics. It just doesn't work. You need quantized
fields. I am not an expert in field theory myself- I am taking my first course
in the subject right now, but I think that this is the biggest tripping point of
students when they learn the theory- why is it that the field equations cannot
be just schroedinger equations.
Ron Maimon