Interval in Quantum Mechanics?

  • #1
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In Special/General Relativity invariance of a space-time interval is just so important. But in Quantum Mechanics, be it non-relativistic or QFT, there seems to be no such parallel. I have always noticed this.
I have some ideas about the reason:

1 - it's not part of the theory to have a conserved interval
2 - there's no way to have a metric in a complex Hilbert space

On the other hand, in QM / QFT conservation of probability seems to be as important as a metric interval is in Special/ General Relativity. So that confuses me.

Probably the answer is that as QM/QFT are worked out in Hilbert Spaces, there's an inner product, which plays the role of a "metric interval" as we know in Special/General Relativity?
 

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  • #2
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be it non-relativistic or QFT, there seems to be no such parallel

In NRQM, there is no role for the spacetime interval because it's non-relativistic. In QFT, everything is Lorentz invariant and interval plays exactly the same role as distance does in non-relativistic QM.

Have you had a course in QFT? Can you work any of the problems?
 
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  • #3
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@kent davidge you are right about the last statement.
In relativity
$$A\cdot B=\sum_{\mu,\nu} g_{\mu\nu}A^{\mu}B^{\nu}, \;\;\; \mu,\nu=0,1,2,3.$$
In QM
$$\langle A| B\rangle=\sum_{i,j} \delta_{ij}A^{i*}B^j, \;\;\; i,j=1,\ldots, {\rm dim}{\cal H}$$
where ##A^i=\langle i|A\rangle##, ##B^j=\langle j|B\rangle##.
 
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  • #4
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Have you had a course in QFT?
I've been going through Weinberg's first of his three volumes in QFT. But it has been some time since I last read the book.
In QFT, everything is Lorentz invariant and interval plays exactly the same role as distance does in non-relativistic QM
But in Special / General Relativity everything can be made Lorentz invariant and space-time interval comes in very explicitaly. :frown:
 
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@kent davidge you are right about the last statement.
In relativity
$$A\cdot B=\sum_{\mu,\nu} g_{\mu\nu}A^{\mu}B^{\nu}, \;\;\; \mu,\nu=0,1,2,3.$$
In QM
$$\langle A| B\rangle=\sum_{i,j} \delta_{ij}A^{i*}B^j, \;\;\; i,j=1,\ldots, {\rm dim}{\cal H}$$
where ##A^i=\langle i|A\rangle##, ##B^j=\langle j|B\rangle##.
Perhaps it is more illuminating to write this as
$$A\cdot B=\sum_k A_kB^k$$
in both relativity and QM. The difference is that in relativity
$$A_k=\sum_l g_{kl}A^l$$
while in QM
$$A_k=(A^{k})^*$$
 
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  • #6
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But in Special / General Relativity everything can be made Lorentz invariant and space-time interval comes in very explicitaly.

It also does in QFT, since QFT requires a Lorentzian background spacetime.
 
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  • #7
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It also does in QFT, since QFT requires a Lorentzian background spacetime.
No it doesn't. Not all QFT's are relativistic QFT's.
 
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  • #8
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While that's true, I don't think those are the kinds of theories the OP is talking about.
 
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  • #9
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While that's true, I don't think those are the kinds of theories the OP is talking about.
Perhaps, but the statement that QFT has Lorentz invariant space-time interval may further confuse the OP.
 
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  • #10
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Thank you to everyone.
 

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