About the partial differential symbol used by F. Duncan M. Haldane.

Hongyi Zhao

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Jul 12, 2021, 9:06:09 PM7/12/21

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Haldane gave a talk on his 2012 ICTP Dirac Medal, of which the
corresponding presentation can be retrieved online from
<https://physics.princeton.edu//~haldane/talks/dirac.pdf>. On the
page 6 of this talk in the above-mentioned file, he wrote the
following formula:

\left|\partial_{\mu} \Psi(\boldsymbol{x})\right\rangle \equiv
\sum_{i} \frac{\partial u_{i}(\boldsymbol{x})}{\partial x^{\mu}}|i\rangle

But I'm confused on the symbols used here. Any more hints/explanations
willl be highly appreciated.

Regards,
HY

Jos Bergervoet

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Jul 13, 2021, 4:49:08 AM7/13/21

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It seems to be factoring of the total wave function
Psi into a spinor part |i> and a spatial part u_i(x),
where the summation index i is then over the number
of spin states.

It is then assumed that the space-time-dependence is
through the u_i(x) and that the |i> are fixed in time.

--
Jos

Hongyi Zhao

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Jul 13, 2021, 11:12:07 PM7/13/21

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On Tuesday, July 13, 2021 at 4:49:08 PM UTC+8, Jos Bergervoet wrote:
> On 21/07/13 3:06 AM, Hongyi Zhao wrote:
> > Haldane gave a talk on his 2012 ICTP Dirac Medal, of which the
> > corresponding presentation can be retrieved online from
> > <https://physics.princeton.edu//~haldane/talks/dirac.pdf>. On the
> > page 6 of this talk in the above-mentioned file, he wrote the
> > following formula:
> >
> > \left|\partial_{\mu} \Psi(\boldsymbol{x})\right\rangle \equiv
> > \sum_{i} \frac{\partial u_{i}(\boldsymbol{x})}{\partial x^{\mu}}|i\rangle
> >
> > But I'm confused on the symbols used here. Any more hints/explanations
> > willl be highly appreciated.
> >
> > Regards,
> > HY
> >
> It seems to be factoring of the total wave function
> Psi into a spinor part |i> and a spatial part u_i(x),

How do you deduce that |i> is a spinor part and u_i(x) is a spatial
part? Based on the talk file given on the website, I can only see
that |i> is a set of fixed orthonormal basis and u_i(x) is the i-th
expanding coefficient of "\Psi x" on this basis.

> where the summation index i is then over the number
> of spin states.

Again, based on the context of the formula, I can not see where the
author speaks of a "spin state".

> It is then assumed that the space-time-dependence is
> through the u_i(x) and that the |i> are fixed in time.

I really can't find this implication too.

Jos Bergervoet

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Jul 14, 2021, 4:39:06 AM7/14/21

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On 21/07/14 5:12 AM, Hongyi Zhao wrote:
> On Tuesday, July 13, 2021 at 4:49:08 PM UTC+8, Jos Bergervoet wrote:
>> On 21/07/13 3:06 AM, Hongyi Zhao wrote:
>>> Haldane gave a talk on his 2012 ICTP Dirac Medal, of which the
>>> corresponding presentation can be retrieved online from
>>> <https://physics.princeton.edu//~haldane/talks/dirac.pdf>. On the
>>> page 6 of this talk in the above-mentioned file, he wrote the
>>> following formula:
>>>
>>> \left|\partial_{\mu} \Psi(\boldsymbol{x})\right\rangle \equiv
>>> \sum_{i} \frac{\partial u_{i}(\boldsymbol{x})}{\partial x^{\mu}}|i\rangle
>>>
>>> But I'm confused on the symbols used here. Any more hints/explanations
>>> willl be highly appreciated.
>>>
>>> Regards,
>>> HY
>>>
>> It seems to be factoring of the total wave function
>> Psi into a spinor part |i> and a spatial part u_i(x),
>
> How do you deduce that |i> is a spinor part and u_i(x) is a spatial
> part?

I did not deduce that it is so with certainty (that is why I
wrote *It seems* at the beginning). To me it seems the most
straightforward interpretation without any further information,
because such a factorization is quite usual.

> Based on the talk file given on the website, I can only see
> that |i> is a set of fixed orthonormal basis and u_i(x) is the i-th
> expanding coefficient of "\Psi x" on this basis.
>
>> where the summation index i is then over the number
>> of spin states.
>
> Again, based on the context of the formula, I can not see where the
> author speaks of a "spin state".
>
>> It is then assumed that the space-time-dependence is
>> through the u_i(x) and that the |i> are fixed in time.
>
> I really can't find this implication too.

Again, this is not an implication, it is in my view a very likely
explanation (and in OP you asked for *any* explanation!) You will
have to look at the rest of the presentation for reasons to believe,
or not believe this explanation.

In any case we can conclude that it is a factorization of the state
space where only the first part contains space-time dependence.
And it contains a sum (as opposed to a single product of two factors)
so it describes entangled states of the two parts. Making it highly
suggestive that this is the splitting of spatial and internal degrees
of freedom. Internal degrees of freedom most likely contain at least
spin. But admittedly, there is no hard proof. Especially these |i>
could contain more than only spin.

One can also wonder why one part, u_i(x), is written as a function,
and the other, |i>, uses Dirac notation. (But given the circumstances
of the presentation, the author may have had other reasons for wanting
to use the latter..)

--
Jos

Hongyi Zhao

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Jul 15, 2021, 2:14:46 AM7/15/21

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On Tuesday, July 13, 2021 at 4:49:08 PM UTC+8, Jos Bergervoet wrote:

The following it the explanation by Haldane himself, and I posted here,
just FYI:

```
partial_{\mu} means \frac{\partial}{\partial x^{\mu}}

(contravariant and covariant index placement is being used)
this is a standard notation in multidimensional differential geometry.
(here the geometry of the parameter space x^{\mu}, \mu = 1,2--..D
```
Regards,
HY