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In Wigner's classification, one observes that the full automorphism group of the Lorentzian manifold $\mathbb{R}^{1,3}$ is precisely the well-known Poincare group. Motivated by this and basic principles of quantum mechanics, its irreducible projective representations are then studied (by passing to unitary representations of the cover).

My question is this: how should I think of the relation between (projective) Hilbert space $\mathcal{H}$ and the actual space-time manifold? If I am in a frame $X$ and someone else is in a different frame $Y$ related to mine via the transformation $\Lambda$, how does this induce a transformation on the Hilbert space? If I am correct, I should not think of both observers having different Hilbert spaces associated to them, but somehow $\Lambda$ gives rise to a projective automorphism of $\mathcal{H}$. I do not see how this happens (except maybe just via the regular representation if the Hilbert space happens to be one of certain functions on $\mathbb{R}^{1,3}$ but this need not be the case if we just consider spin for example.

Maybe this is a silly question, but I hope I managed to convey my confusion.

Thomas Bakx
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    I'm not completely clear on your confusion. Is it that the Hilbert space is defined for a particular time slice, and a generic element of the Poincare group (take e.g. a boost) does not preserve the time slicing? – Logan M May 02 '18 at 15:26
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    Hi, thanks for getting back to me. I'm afraid I am confused about what exactly it is that I am confused about. Given a physical system, I just want to know what the relation is between its Hilbert space (or state space for that matter) and the actual spacetime it lives in. I want to understand Wigner's motives for considering the action of the Poincare group on the state space, not the space-time itself (of which it really is the symmetry group). I hope this helps. – Thomas Bakx May 02 '18 at 15:59
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    @Thomas Bakx: Being confused about what you're being confused about is a common feeling when you're studying QM/QFT! – Mozibur Ullah May 03 '18 at 12:56

2 Answers2

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As I understand things, vectors (kets) that stand for particles are in the very large dimensional Hilbert space. The Hilbert space is not "space-time". Representations of the Poincare group act on the kets in the Hilbert space to rotate, boost, and translate the kets by the Lie group parameters $\vec{\theta},\vec{\lambda},\vec{x},t$. The Poincare group acts on its own generators by conjugation such that under its rotation subgroup, $\vec{\theta},\vec{\lambda},\vec{x}$ rotate like 3-vectors, and under its Lorentz subgroup, $(\vec{x},t)$ transforms like a 4-vector. It is the space of the Lie Group parameters $(\vec{x},t)$ that you call "space-time".

Gary Godfrey
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  • Thanks! You say that the Poincare group 'acts on the kets to rotate, boost or translate the kets'. Can you give a concrete example of such an action in a physical context (i.e. where it is related to an experiment performed by observers in two different Lorentz frames)? – Thomas Bakx May 03 '18 at 08:02
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How should I think about the relation of the Hilbert space to spacetime?

This was something of a puzzle for me when I first learnt about QM. In classical mechanics as well as SR & GR everything is located within space and time.

The Hilbert space of states supervenes on spacetime. This is a philosophical term that means 'a fact or a property is entailed or consequent on the existence or establishment of another.' Generally the former is a higher level phenomenon and the latter at a lower level.

In fact, the mathematical structure and physics of QM is silent on where this Hilbert space is located. It's not anywhere. This is an ontological puzzle that puts the speculation about higher spatial dimensions in string theory into the shade. At least there, we know what it is we are talking about, even if we have no direct experimental verification of such a hypothesis.

If I am in a frame X and someone else is in a frame Y, how is the Hilbert space at X related to that of Y?

I'm not sure on this one. It's a good question. What I can say is that in canonical quantisation in QFT equal time commutation rules are applied on momenta & position at different locations in space but at the same time. This however, seems to break the relativity, as there the concept and possibility of simultaneity is denied.

Mozibur Ullah
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  • Thanks! Do you have any sources that can perhaps shed more light on this matter? I am trying to find a concrete example, as I commented on the other answer. – Thomas Bakx May 03 '18 at 08:08
  • @Thomas Bakx: I'm not sure that I can; it's something that I've understood by reading various QM texts; I think on the whole, these texts could be a lot clearer about such simple, yet fundamental questions - which is why it's difficult to recommend anything; if I think of one, I will let you know. – Mozibur Ullah May 03 '18 at 12:51