The idea of impulse, i.e. Change in momentum = Integral dt Force appears in Newtonian mechanics. Impulse is considered to be a physical observable, for example it may be linked to the damage done by a particle hitting a target. Impulse, however, only depends on momentum and so two particles with the same momentum, but different masses and velocities may cause the same damage i.e. represent the same physical observable. Momentum in nonrelativistic Newtonian mechanics is m v where m is rest mass and v velocity. Velocity requires a clock in order to measure it, yet two particles with the same momentum and different velocities are considered equivalent in terms of the damage done upon collision with a target i.e the physical impulse. Thus we suggest that time must formally disappear from mv, but this implies that rest mass must behave as t or dt. Furthermore, given Kinetic energy = .5mvv, it must behave as 1/dt. In other words, the reciprocal of kinetic energy behaves as a gradation of time. This is not a standard clock as the gradation is based on a specific kinetic energy and is different for another. In other words, two particles with the same p and different v, have different time gradations. This, however, is a quantum mechanical result. Thus we argue that quantum mechanics may be responsible for the idea of p yielding the same impulse hit for two different v. We further argue that these ideas may be extended to include rest mass with v=0 and to indicate that 1/p is linked to dx, a ruler gradation which is again a quantum mechanical result. Lastly, we argue that these ideas lead to the special relativistic result EE = ppcc + mmcccc and so argue that free particle quantum mechanics and special relativity are closely linked and that classical impulse follows from free particle quantum mechanics.
The idea of impulse, i.e. Change in momentum = Integral dt Force appears in Newtonian mechanics. Impulse is considered to be a physical observable, for example it may be linked to the damage done by a particle hitting a target. Impulse, however, only depends on momentum and so two particles with the same momentum, but different masses and velocities may cause the same damage i.e. represent the same physical observable. Momentum in nonrelativistic Newtonian mechanics is m v where m is rest mass and v velocity. Velocity requires a clock in order to measure it, yet two particles with the same momentum and different velocities are considered equivalent in terms of the damage done upon collision with a target i.e the physical impulse. Thus we suggest that time must formally disappear from mv, but this implies that rest mass must behave as t or dt. Furthermore, given Kinetic energy = .5mvv, it must behave as 1/dt. In other words, the reciprocal of kinetic energy behaves as a gradation of time. This is not a standard clock as the gradation is based on a specific kinetic energy and is different for another. In other words, two particles with the same p and different v, have different time gradations. This, however, is a quantum mechanical result. Thus we argue that quantum mechanics may be responsible for the idea of p yielding the same impulse hit for two different v. We further argue that these ideas may be extended to include rest mass with v=0 and to indicate that 1/p is linked to dx, a ruler gradation which is again a quantum mechanical result. Lastly, we argue that these ideas lead to the special relativistic result EE = ppcc + mmcccc and so argue that free particle quantum mechanics and special relativity are closely linked and that classical impulse follows from free particle quantum mechanics.
In Newtonian mechanics, one often thinks in terms of x(t) i.e. an association between x and t. Although this is part of Newtonian mechanics, the notion of motion always being linked to time (and so an internal clock being required) is not always the case. For example, two particles with different masses and different velocities such that their momenta are the same may strike a target and reflect with the same magnitude of momenta. The impulse is 2p for both even though they have different speeds. One does not need to consider their speed to discuss the impulse change and so an external clock is not needed. This is further emphasized by the Lorentz invariant A = -Et+px which separates x and t as if they were independent variables (even though implicitly a speed exists in the problem i.e. v=dx/dt.). The point is that there exists a subset of physics (i.e. interactions) which is not dependent on velocity and hence the presence of a clock. The impulse hit discussed above is one example, but interference through 2 slits etc is another. Two quantum particles with the same momentum, but different rest masses and velocities produce the same interference patterns. A particle with momentum, however, has energy and so both time and space are running variables, but there are problems in which one considers p without a clock. Interestingly, even if one thinks in terms of impulses without using a clock, if there is an x interval between impulses then the notion of velocity comes into play even without an external clock because there will be more impulses imparted by a fixed p with a higher velocity than a lower one in the same fixed time period. Without using the idea of a clock, this give rise to the observable called pressure in classical physics, but is also present in a quantum bound state, we argue. Pressure in classical physics is proportional to temperature which in turn is proportional to energy (and a running time). In a quantum bound state, the equivalent to pressure (namely impulse hits with a potential) also require an energy viewpoint which is linked to running time i..e exp(-i En t) where En is the bound energy.
In classical physics, velocity follows from exact measurements of space and time intervals (i.e perfect resolution of both). Inertial mass follows from a resistance of an object to a force (push or pull). Momentum, however, which nonrelativistically is mv and should be associated with ideas of inertial mass, and space time resolution may be instead linked to the physical idea of an impulse. An impulse seems to be like a hit and does not appear to be linked directly to space time physically, although its consequences are i.e. a particle receiving the impulse moves in space and time. In classical mechanics, x(t) and so one considers either equally spaced dt intervals and corresponding dx(t) or vice versa (constant dx) if an integral only involves space dependent values. In quantum mechanics, one has two uncertainty principles, one related to E and t and the other to p and x. We argue that p is linked to the notion of an impulse which does not on the surface involve space or time. Given that p, even in nonrelativistic quantum mechanics, is mv there should at least be some built in notion of dx and dt. From the uncertainty principles one sees that dx (i.e. the wavelength) is linked to hbar/p and dt to hbar/E. Thus neither dt nor dx are constants, but depend on the motion. In the high energy quantum limit dt —> W(x)W(x) dx →1/v(x) dx where v(x) is classical velocity and W(x) the bound state wavefunction. The bound state wavefunction, however, applies to the envelope of WW (not the actual crests and troughs) and so the dx-dt scheme which emerges is a smoothed over one and not the actual dx -dp of quantum mechanics. We try to examine some of these ideas in this note.
Classical Newtonian mechanics separates a particle (effect) from a force (cause). A particle has properties at a given x (and time t), but if it interacts with a force, this interaction occurs over dt and dx. Given that many forces do not change in time, we argue that a notion of a force is inseparable from the idea of dx. If one gives the value of a force at x, this value acts within a dx, which in Newtonian mechanics tends to 0. We have argued before that a particle with momentum represents a force or impulse itself, but at the same time may receive impulse hits from other sources (e.g. a Newtonian force -dV/dx). If a particle represents a force(impulse) it should be inseparable from the notion of dx, even if it has a constant p (momentum) and is traveling through space. Thus the idea of a momentum defining an internal ruler gradation through a probability distribution in x which we have presented in previous notes is compatible with the idea of describing a particle as a force. If one only considers a particle as an effect, there would be no need to worry about dx until a force(x) appears. An internal dx is not defined by an external ruler and does not need to tend to 0 as in Newtonian mechanics. Rather it is statistical in nature as p is a fixed impulse, but the probability to be at different x points within dx (called a wavelength) differ. In (1) we compared a classical Maxwell-Boltzmann gas to a quantum free particle. We argued that d/d density Pressure = RT = average pressure of a particle. We suggested this is analogous to the quantum free particle situation: (-id/dx probability(p,x) ) / probability(p,x) = p where probability(p,x) is unnormalized. In other words a change in a probability yields a deterministic value p. The notion of a deterministic value p associated with a probability distribution in what would be another deterministic variable x in Newtonian mechanics (except now one has a dx distribution because the particle is a force/impulse) has consequences. If one wishes to have a particle sit at x=0, say in an oscillator system, one needs to use a wavelength which is fairly small and centered at x=0. The wavelength, however, is associated with momentum and even being small spreads out from x=0 meaning there is potential energy .5kxx as well. Thus one may use an ensemble of forces/impulses Sum over p a(p)exp(ipx) to create a “dx” about x=0, but at the same time this ensemble represents an average particle (defined at x with prms(x)) which must satisfy KE(x) + V(x) = En i.e. the classical conservation expression. In other words, one sees again that average statistical quantities may behave as deterministic ones.
Classical mechanics defines the concepts of work = Integral dx F = change in kinetic energy and impulse= Integral dt F = change in momentum (vector). We argue that these two concepts are associated with different physical processes and contain different information. For example, a change in momentum is associated with an impulse e.g. the striking (and possibly breaking of a target) and there is a degeneracy in that p=m1v1=m2v2. Kinetic energy, on the other hand, describes a different physical process, namely the “climbing of a potential hill” and one may have different (mo, p) sets yielding the same kinetic energy, just as the same p (same impulse) may be associated with different kinetic energies. Thus there are two distinct physical processes (impulse and interchange of kinetic energy or energy) and each of these are associated with different information. In quantum mechanics, exp(-iEt) describes the time dependent part of a free particle wavefunction (special probability) while exp(ipx) describes spatial. We have argued in previous notes that this probability should be associated with entropy, but in that case there should be an entropy in time and one in space. Given that E and p differ numerically, one needs a reason for the existence of two different entropies. We argue that they are associated with different physical processes and information and so two different entropies may be possible. (We use nonrelativistic expressions here, but the same ideas apply to the special relativistic case.)
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
Historically, special relativity followed as a generalization of Newtonian mechanics and later quantum mechanics (both nonrelativistic and relativistic) was developed as a somewhat independent theory with some connections to Newtonian/special relativistic concepts such as energy and momentum and their conservation. For example, the Newtonian concept of a potential V(x) is used in quantum mechanics. Here we wish to consider the notion of a system seen from different frames moving with constant speed (-v) together with some basic physical ideas that are presumed to hold at any scale such as the idea of a push or pull changing the speed of a particle and the notion of speed itself. We argue that one may develop special relativistic equations in such a manner, but at the same time consider notions of dp/dt and dE/dx linked not only to Newtonian mechanical ideas, but at the same time to quantum mechanics, by generalizing these forms. We argue that one may start with the notion of energy, momentum and their conservation together with the generalizations of a linear force linked to finite dt=hbar/E and dx=hbar/p. This allows one to a priori write, based on conservation of energy for v<