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In other words, is Minkowski space a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product? I would think that Minkowski space can either be thought of as 2-D complex or 4-D real, do either of these interpretations change whether or not it is a Hilbert space?

José Carlos Santos
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wildturtle
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  • See https://math.stackexchange.com/questions/526171/is-minkowski-space-not-a-metric-space – cmk Jul 02 '19 at 17:46

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A Hilbert space is a real or complex vector space with an inner product which induces a norm under which the space is complete. Minkowski space incorporates three dimensional space together with a time dimension. It comes with a Lorentzian inner product which is not an inner product (it is not positive definite). When one says 'Minkowski space' one usually means the geometry of the space with the Lorentzian inner product. Therefore, no, Minkowski space is not a Hilbert space.

Ittay Weiss
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No. In Minkowski space we have a bilinear form$$\bigl\langle(t,x,y,z),(t',x',y',z')\bigr\rangle=-tt'+xx'+yy'+zz'$$which is not an inner product. Therefore, it doesn't induce a metric.

José Carlos Santos
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  • So you assume that Minkowski space is $\mathbb R^4$? – Filippo Jun 16 '22 at 11:21
  • @Filippo Indeed. As the Wikipedia article about the Minkowski space says, “Minkowski space (…) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.” – José Carlos Santos Jun 16 '22 at 11:30
  • In other words, you are saying that an event in spacetime - i.e. an element of Minkowski space $M$ - is a list of $4$ real numbers. Well, we can use a Minkowski chart to assign $4$ real numbers to each event (since a Minkowski chart is bijection from $M$ to $\mathbb R^4$), but there is no preferred chart and thus I wouldn't say that Minkowski space is $\mathbb R^4$. But I am open to a discussion. – Filippo Jun 16 '22 at 11:49
  • @Filippo Well, I am not. I suggest that you post a question concerning this. – José Carlos Santos Jun 16 '22 at 11:59