it all depends i reckon how we define 'god'. what do we mean by 'higher' dimensions'? why not 'deeper' dimensions'. The 'deeper' would be sympathetic with pre-mechanisltic understandings of Nature and us. For example the pagan belief of the 'Underworld' and/or 'Therworld' a timeless dimension(s) that coexists with ours, and that apparently can be contacted in various states of non-ordinary awareness
OK, with that in mind, let's pretend we ask a person who is with Earth religion, eg a Dionysian celebrant what 'god' is and what 'higher dimensions' is--how would s/he answer?.....And how about a person who follows Abrahamic religion? How would they answer what 'God' is and what 'higher dimension(s) is?
For the former, maybe they would understand 'god' to be what they beCOME in non-ordinary awaress, DIRECTLY experiencing 'deeper or higher dimension(s).
Whereas the Abrahamite may claim that 'God' is transcendental IN a higher dimension. For s/he non-ordinary experience OF any sense of a 'higher or deeper dimension' is forbidden by their authorities.
And physics' 'myth' is devoid of poetry that speaks to such an interelationsip in an orgiastic way. What do you say?