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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

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

Jul 13, 2021, 4:49:08 AM7/13/21

to

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

Jul 13, 2021, 11:12:07 PM7/13/21

to

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
> 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),

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.

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.

Jul 14, 2021, 4:39:06 AM7/14/21

to

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
> 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?

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.

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

Jul 15, 2021, 2:14:46 AM7/15/21

to

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

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

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