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Author: Jay R. YablonJay R. Yablon Date: May 10, 2008 17:28
To all,
I have several queries about the Dirac equation and associated Dirac
spinors psi and quantum field theory (QFT).
1. In units hbar=c=1, a Dirac spinor psi has mass dimension +3/2. Yet,
the "inside" of a Dirac spinor for, say, a spin up electron is often
written as (transposed, easier for ASCII):
(1 0 p_z/(E+m) P_+/(E+m) ) (1)
which is dimensionless, and the normalization factor is written as:
sqrt [(E+m)/2m] or sqrt(E+m) (2)
The latter of (2) at least has mass dimension of 1/2, but what is the
representation (or normalization) of this which explicitly shows the
+3/2 mass dimensionality? Do we just multiply (1) through by m? Or by
E+m? What is the normalization for psi*T psi with mass dimension +3
which drives this? *T=conjugate transpose.
2. The Dirac spinor in compact form is often written as:
psi = u(p) exp [-i p x] (3)
I understand what happens from there, but why start with a complex plane
wave? Why not, for example, start with a more general form:
psi = u(p) exp [-(1/2)Ax^2 + Bx + V(x)] (4)
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Author: Oh NoOh No Date: May 11, 2008 01:10
Thus spake Jay R. Yablon nycap.rr.com>>To all,
>
>I have several queries about the Dirac equation and associated Dirac
>spinors psi and quantum field theory (QFT).
>
>1. In units hbar=c=1, a Dirac spinor psi has mass dimension +3/2. Yet,
>the "inside" of a Dirac spinor for, say, a spin up electron is often
>written as (transposed, easier for ASCII):
>
>(1 0 p_z/(E+m) P_+/(E+m) ) (1)
>
>which is dimensionless, and the normalization factor is written as:
>
>sqrt [(E+m)/2m] or sqrt(E+m) (2)
>
>The latter of (2) at least has mass dimension of 1/2, but what is the
>representation (or normalization) of this which explicitly shows the
>+3/2 mass dimensionality? Do we just multiply (1) through by m? Or by
>E+m? What is the normalization for psi*T psi with mass dimension +3 ...
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Author: Jay R. YablonJay R. Yablon Date: May 11, 2008 07:11
> Thus spake Jay R. Yablon nycap.rr.com>>>To all,
>>
>>I have several queries about the Dirac equation and associated Dirac
>>spinors psi and quantum field theory (QFT).
>>
>>1. In units hbar=c=1, a Dirac spinor psi has mass dimension +3/2.
>>Yet,
>>the "inside" of a Dirac spinor for, say, a spin up electron is often
>>written as (transposed, easier for ASCII):
>>
>>(1 0 p_z/(E+m) P_+/(E+m) ) (1)
>>
>>which is dimensionless, and the normalization factor is written as:
>>
>>sqrt [(E+m)/2m] or sqrt(E+m) (2)
>>
>>The latter of (2) at least has mass dimension of 1/2, but what is the ...
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Author: Oh NoOh No Date: May 11, 2008 09:41
Thus spake Jay R. Yablon nycap.rr.com>>> Thus spake Jay R. Yablon nycap.rr.com>>>>To all,
>>>
>>>I have several queries about the Dirac equation and associated Dirac
>>>spinors psi and quantum field theory (QFT).
>>>
>>>1. In units hbar=c=1, a Dirac spinor psi has mass dimension +3/2.
>>>Yet,
>>>the "inside" of a Dirac spinor for, say, a spin up electron is often
>>>written as (transposed, easier for ASCII):
>>>
>>>(1 0 p_z/(E+m) P_+/(E+m) ) (1)
>>>
>>>which is dimensionless, and the normalization factor is written as:
>>>
>>>sqrt [(E+m)/2m] or sqrt(E+m) (2) ...
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Author: Jay R. YablonJay R. Yablon Date: May 11, 2008 17:40
> Thus spake Jay R. Yablon nycap.rr.com>>>> Thus spake Jay R. Yablon nycap.rr.com>>>>>To all,
>>>>
>>>>I have several queries about the Dirac equation and associated Dirac
>>>>spinors psi and quantum field theory (QFT).
>>>>
>>>>1. In units hbar=c=1, a Dirac spinor psi has mass dimension +3/2.
>>>>Yet,
>>>>the "inside" of a Dirac spinor for, say, a spin up electron is often
>>>>written as (transposed, easier for ASCII):
>>>>
>>>>(1 0 p_z/(E+m) P_+/(E+m) ) (1)
>>>> ...
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Author: Oh NoOh No Date: May 12, 2008 00:13
Thus spake Jay R. Yablon nycap.rr.com>>
>>>>
>>>>> Why not, for example, start with a more general form:
>>>>>
>>>>>psi = u(p) exp [-(1/2)Ax^2 + Bx + V(x)] (4)
>>>>>
>>>>>where V(x) is a general polynomial in x?
>>>>
>>>> It is not known how to construct interacting quantum fields. Haag's
>>>> theorem causes me to believe this cannot be done. If V is a
>>>>polynomial
>>>> in x, why do you need the A and B terms. (4) does not seem to me
>>>>more
>>>> general, because, a) it is real and in general the wave function is
>>>> complex and b) plane wave states are a basis (in loose terms), which
>>>> means they give a completely general solution.
>>>
>>>A and B can be made real or imaginary by substituting A-->iA, B-->iB.
>>>There is no reason why NOT to have the A, B terms, and V could, in ...
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Author: Jay R. YablonJay R. Yablon Date: May 12, 2008 08:07
> Thus spake Jay R. Yablon nycap.rr.com> . . .
>>>>> It seems to me that if you start with (4), you would be solving
>>>>> the
>>>>> interacting Dirac equation.
>>>>
>>>>I am certainly happy to be doing that.;-)
>>>>
>>>
>>> Even though there is no way from there to quantum field theory?
>>
>>Actually, now that we have discussed this, I believe there is a way.
>>It will take several days for me to write it up; tonight I want to do
>>the calculations to see if it hangs together the way I think it will.
>>Basically, using:
>>
>>psi = u(p) e^psi = u(p) exp [-(1/2)Ax^2 + Bx + V(x)] (A)
>> ...
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Author: Oh NoOh No Date: May 12, 2008 13:00
Thus spake Jay R. Yablon nycap.rr.com>>> Using Gaussians makes no sense to me. Wave functions in general are
>> superpositions. This is not even an orthornormal set of states.
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