Interval in Quantum Mechanics?

[kent davidge]

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?

 

[Vanadium 50]

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?

 

[Demystifier]

@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##.

 

[kent davidge]

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.

 

[Demystifier]

@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
$$A_k=(A^{k})^*$$

 

[PeterDonis]

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.

 

[Demystifier]

It also does in QFT, since QFT requires a Lorentzian background spacetime.

No it doesn't. Not all QFT's are relativistic QFT's.

 

[Vanadium 50]

While that's true, I don't think those are the kinds of theories the OP is talking about.

 

[Demystifier]

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.

 

[kent davidge]

Thank you to everyone.