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Independence of Time and Space In Quantum Mechanics Francesco R. Ruggeri Hanwell, N.B. Nov. 26, 2025 In (1), it is stated that “time flows continuously like a river’ … and is treated like a classical variable in Quantum Mechanics.” Here, we wish to consider the behaviour of time in quantum situations. We argue that a 4-space sense of time appears in quantum mechanics through exp(-iEt+ipx), but only intertwined with E (energy) and p (momentum), i.e. interactions involving these variables. Furthermore, there suggest that there must be a clear measure of t as well as an interaction changing E to use exp(-iEt) and a clear measure of x and an interaction changing p to use exp(ipx). In particular, we note that the classical equation x/t = v ensures that time and x are not on the same footing and this same equation holds in special relativity, so -Et+px = constant ((1)) which has the appearance of x and t being on the same footing (as well as (x,ct)) is subject to x/t=v (for xo=0, to=0). Thus, if x/t=v governs x and t information, then x and t are not independent. It seems that one would require fluctuations from these values. The fluctuations ranges could be linked, but actual t and values within the fluctuations would be independent. . We suggest it is possible to find a formalism which treats x and t independently from ((1)), but this formalism must be intertwined with E and p. In particular, one has free particle quantum mechanics in which delta t = hbar/E and delta x = hbar/p. X and t are decoupled in this scenario and so appear as independent because there is no notion of x/t=v when dealing with quantum type interactions. When dealing with such interactions, one uses the probability exp(-iEt+ipx), but this begs the question: How is this factor really used? We suggest that one must consider the following. Given exp(ipx), one uses it if p changes and there is a distinct measure of changing x in the problem. This scenario applies to two-slit interference, bound states with V(x) and one dimensional reflection-refraction at a point x=0. Given exp(-iEt), one uses it if there is a distinct interaction involving E, i.e. it changes and there is a distinct measure in time. Thus, two-slit interference, one dimensional reflection-refraction and elastic scattering from a V(x) are out because they don’t involve changes in E. A bound state with V(x) certainly involves changes in E, but does not have a direct measure of time t (independent of x). A bound state with V(x,t), however, does have a direct measure of t (being independent of x) and also involves energy changing and so exp(-iEt) may be used in this case, as will be shown. Special Relativity, Newtonian Mechanics and Time Newtonian mechanics seeks to obtain x(t) which shows that x (or y, z for that matter) are not independent of time. There is no sense of 4-space in Newtonian mechanics. 4-space appears in special relativity because the Lorentz transform acts on the 4-vector (x,y,z, ct) which is seemingly independent in the four variables, but really is not. To see this, consider the simpler 2-dimensional case: | g(v) v/c g(v) | |x| ((2)) g(v) = 1/sqrt(1-vv/cc) |v/cg(v) g(v) | |ct|
The vector (x,ct) appears to treat x and ct as independent variables, but the presence of: v= x/t for xo=0, to=0 ((3)) The matrix of ((2)) shows that this is not the case. For x and t to be independent, one cannot use an equation which treats them differently as ((3)) does. Thus, x and t are not independent in special relativity. Special Relativity and Free Particle Quantum Mechanics Even though x/t=v holds for special relativity, we suggest that using the Lorentz invariant: A = -Et+px (= classical relativistic and nonrelativistic action with x/t=v) ((4)) allows one to define uncertain regions: Delta x = hbar/p and delta t = hbar/E ((5)) It is within these uncertainty relations that x and t become independent and these are directly linked to an interaction. Thus, x=vt still holds on average and x and t are not independent in this case, but this has nothing to do with an interaction. The interaction is described by: exp(ipx) exp(-iEt) ((6)) ((6)) is not linked to x=vt and two-slit interference and 1-dimensional reflection-refraction at an n1--n2 index of refraction junction show that there is non-classical physics present. We suggest that this non-classical physics involves independence of t and x, just as ((5)) are independent. As a result, x and t are taken to be independent in ((6)). This begs the question: How should one use ((6))? How To Use exp(-iEt+ipx) At first, it seems that one would use exp(-iEt+ipx) in a classical manner, applying x and t values subject to x=vt. We suggest, however, that exp(-iEt+ipx) shows that interactions in t and x are linked with probability regions associated with both variables (i..e ((5))). Thus, a p, which delivers an impulse hit does not act at a single x point, but there is a probability range of hbar/p. Thus, a particle may interact with both slits of a 2-slit apparatus if they are about hbar/p apart. This means that x is not certain. We argue that hbar/E means that time is not certain either and so does not really flow like a river when interactions occur. The question then becomes: How does one apply exp(-iEt+ipx). We suggest the following scheme:
((7a)) exp(ipx) is used if p changes in an interaction and there is measure of x which shows how it may change ((7b)) exp(-iEt) is used if an interaction changes E and there is a measure which shows how t changes, independent of x. The scheme of ((7)) involves both x,t and p,E and must involve interactions. A free particle moving in space follows x=vt. It is when it interacts that one must use ((7)) because one does not have Newtonian interaction at a point x,t. Newtonian mechanics, however, works very well in the classical world, but this is because hbar/p and hbar/E are almost 0, giving the sense of interaction at a point x,t. In situations in which ((5)) are not 0, one must reject interaction at an x,t. We now consider a series of examples. Examples of exp(-iEt+ipx) In ((7b)), we argued that the use of exp(-iEt) must involve an interaction which changes E. Thus, 2-slit interference calculations, which do not change the magnitude of p, elastic scattering from V(x) and one dimensional reflection-refraction from an n1-n2 index of refraction junction at x=0 are not linked with exp(-iEt). One must use exp(ipx) for these, because p changes in all and there is a measure of x in all of these problems, i.e. the separation of the slits, the n1-n2 junction at x=0 and V(x) in the elastic scattering. We next consider a single particle quantum bound state in V(x). In such a case, there are hits to exp(ipx) which change p and there is a direct measure of x because of V(x). Thus, exp(ipx) may be used. On the other hand, there is a change in energy certainly because as p changes so does E, but there is no measure in time independent of x and so one does not use exp(-iEt). A bound state calculation yields an E value and this may be applied to an overall exp(-iEt), but this is only used if ((7b)) applies, we argue. Finally, we consider V(x,t). This treats x and t as independent. One may fix x in which case there is no measure in x, only in t. V(x,t) represents changes in energy and so ((7b)) applies. Thus, we argue that one should be able to write: W(x=x1, t) = Sum over a(p,x) exp(-i pp/2m t) (nonrelativistic) ((8)) This should lead to: -1/2m d/dx d/dx Sum over p, a(p,x) exp(-ipp/2m t) + V(x1,t) W(x1,t) = E W(x1,t) ((9)) Using: V(x1,t) = Sum over b V(b,x1) exp(ibt) one has: -1/2m d/dx d/dx a(p,x1) + Sum over V(b,x1) a(pp2/m-b) = E a(p, x1) ((10)) For a component p, i.e. pp/2m. (We assume a(p,x1)=a(-p,x1).)
Unlike the x case, one cannot localize in time and so there is no W(x1, t→ infinite) = 0. Conclusion In this note, we try to consider the relationship of x and t. Classically x/t=v, meaning that even in special relativity these variables cannot be independent despite the form (x,ct) and -Et+px = constant. The reason for the lack of independence is the presence of v in the Lorentz transform matrix which implies v=x/t for xo=0 and to=0. As long as x/t=v is present, one does not have independence of x and t. We suggest, however, that there is a case in which they become independent and that is free particle quantum mechanics which we argue follows from A = -Et+px. Even though one may consider an x and t from x/t=v, one has delta x = hbar/p and delta t = hbar/E. It is x and t values within these ranges which are independent of each other and x=vt, even though the lengths of the ranges are linked through 0= -E delta t + p delta x. During an interaction, one may have independent probabilistic changes of x and t for a given p and E which are interacting. We suggest that this is why a particle interferes at a 2-slit apparatus with slits separated about hbar/p. It interacts with both slits. hbar/p and bhar/E lead to the probability exp(-iEt+ipx) and we try to provide rules for using this function. We suggest that x and t are independent in this form. exp(ipx) is used (without exp(-iEt), if p changes and there is a distinct measure of x in the problem independent of t. This applies to 2-slit calculations, elastic scattering from V(x) and 1-D reflection-refraction from an n1-n2 index of refraction junction at x=0. In the case of a single particle bound state in V(x), there is a clear measure of x and p changes. In the case of exp(-iEt), there is also an interaction which changes E, but no measure for t independent of x . In the case of V(x,t) for x=x1, however, there is an interaction which changes E and a clear measure of t independent of x, in other words, one does not introduce a measure in time by replacing x through x=vt. References 1. Connerade, J.-P. The Arrow of Time in Quantum Theory (Oct. 2025) https://www.mdpi.com/2218-2004/13/11/86
