# Why does a phase shift in a light pulse imply a non-commutative structure of space (which implies gravity has a quantum structure)?

In the news report Physicists propose test of quantum gravity using current technology (Lisa Zyga, Phys.org, 27 October 2017), a test is proposed to determine if gravity has a quantum structure. From the report:

With their proposed test, the physicists' goal is to find experimental evidence supporting the idea that spacetime does indeed have a non-commutative structure. To do this, the proposed test attempts to detect any changes in the conventional commutative relations occurring in a micro-mechanical oscillator. If these changes are present, they would indicate a non-commutative structure and produce a measurable optical phase shift on a light pulse that has been coupled to the oscillator.

Can anyone explain why a phase shift of the light pulse implies a non-commutative space structure? Is the outcome of the test already known?

Quantum mechanics, in its most basic form, relies on the fact that position and momentum don't commute, $$[x_i,p_j] = i\hbar\delta_{ij}$$ (where $$x_i$$ and $$p_j$$ are the $$i$$th and $$j$$th components of position and momentum), which in essence encodes the de Broglie relation and the Heisenberg uncertainty principle, and the fact that position and momentum are incompatible observables.

Certain theories of quantum gravity include modifications to spacetime that make it 'noncommutative', which means that there are nonzero commutators $$[x_i,x_j]$$ and $$[p_i,p_j]$$ (as well as modifications to $$[x_i,p_j]$$), so that the different components of position and momentum are no longer compatible observables.

This is a fairly drastic change to quantum theory, and it is easy to see how such modified commutation relations would impact the predictions. The paper in question (Nucl. Phys. B 924 578-587 (2017), arXiv:1710.03920) works through the impacts of such a modification and shows that it is (at least in principle) observable using optomechanical observables.

Can anyone explain why a phase shift of the light pulse implies a non-commutative space structure?

It doesn't. It provides an experimental test that can reveal configurations which are incompatible with standard quantum mechanics and which could be explained with a specific modification thereof. (Assuming, of course, that experiments were indeed to coincide with the authors' predictions and fall outside the range explainable with standard QM, which is not the case yet.)

Such an experiment would be the first piece of evidence towards establishing the theoretical framework used by the authors, but it would require a mountain of additional evidence to "imply" a non-commutative space structure.

• I completely forgot to thank for the effort you took, so: T(h)anks! – Deschele Schilder Aug 21 '20 at 18:35
• One more thing. Certain theories of quantum gravity include modifications to spacetime that make it 'noncommutative' So even if non-commutativity has been proven not to exist, there is still the possibility that quantum gravity might exist? – Deschele Schilder Aug 21 '20 at 18:39
• To be clear, non-commutativity has not been proven not to exist - the paper in question makes no such claims. As to the existence of other theories of quantum gravity that don't use that structure, you'll have to ask separately. – Emilio Pisanty Aug 22 '20 at 11:46
• What do I have to ask separately? It's clear that if non-commutativity has been proven not to exist there will be only quantum gravity theories left playing in the game without a non-commutative structure of spacetime. – Deschele Schilder Aug 22 '20 at 11:52
• As for 'effort', this type of of analysis is not particularly complicated and can be completed quickly, if the question is posed properly. Most of the work in this thread was caused by your lack of fire diligence and your refusal to remove harmful content from your post. I hope you can keep that on mind for future threads. – Emilio Pisanty Aug 22 '20 at 12:05