Anamitra Palit
Registered Senior Member
Consider the following transformation of an arbitrary well behaved function f(x,y), the transformation being given by:
$$F(E,p,x,t)=\int_{-\infty}^{x}\int_{-\infty}^{t}f(x,t)e^{ia(Et-px)}dxdt$$
$$\frac{\partial F}{\partial x}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial x}e^{ia(Et-px)}-f(x,t)iap e^{ia(Et-px)}]dxdt$$
$$\frac{\partial F}{\partial x}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial x}-f(x,t)iap )e^{ia(Et-px)}]dxdt$$----(1)
Similarly,
$$\frac{\partial F}{\partial t}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial t}e^{ia(Et-px)}+f(x,t)iaE e^{ia(Et-px)}]dxdt$$
Or,
$$\frac{\partial F}{\partial t}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial t}+f(x,t)iaE )e^{ia(Et-px)}]dxdt$$----(2)
If $$\frac{\partial F}{\partial x}=0$$ we have from (1) as an option:
$$\frac{\partial f}{\partial x}-f(x,t)iap =0$$
or,$$f(x,t)=Ae^{iapx}$$---- (3)
Again if $$\frac{\partial F}{\partial t}=0$$
then,
$$\frac{\partial f}{\partial t}+f(x,t)iaE=0$$
Or,$$f(x,t)=Be^{-iaEt}$$ ------------- (4)
We may write,
$$f(x,t)=Ce^{-ia(Et-px)}$$ -------------- (5)
And proceed as in the previous manner.
Inserting (5) into(1) we have:
$$F(E,p)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}Ce^{-ia(Et-px)}e^{ia(E_0t-p_0x)}dxdt$$
Or,
$$F(E,p)=Const \times \del (E-E_0)\del (p-p_0)$$ ------------ (6)
We now consider,
$$\psi (x,t)=\Sigma C_ie^{-ia(E_it-p_ix)}$$ ----------- (7)
By inserting (7) into (1) we have,
$$F(E,p)= \Sigma C_i\del (E_i-E_0)\del (p_i-p_0)$$ --------(8)
F(E,p) is the "Transformation " of $$\psi (x,t)$$, the transformation being defined on the first line of the post.It becomes athe Fourier Transform if x and t are allowed to tend to infinity.
F(E,p) if integrated over the (E,p) domain , it counts the number of modes having $$E=E_0$$ and $$p=p_0$$
If the number of such nodes corresponding to some $$(E_0,p_0)$$ is divided by the total number of possible modes we get a probability picture in an expected manner.By Fourier inversion we should get the same picture on the (x,t) domain.
The periodic nature of psi is clear from the expression (7) or (5).Psi is indeed a "wave function" By using suitable boundary conditions we restrict ourselves to discrete values of E and p.
Observations:
1. Equations (5) and (7) satisfy the Klein Gordon Equations.
2 Invariance of the exponent in (5) or (7) may be achieved through the Lorentz transformations provide "a" is an invariant. This corresponds to the fact that $$\hbar$$ is an universal constant.
3.If x and t represent represent x-coordinate(spatial) and time respectively then momentum and energy would be suitable candidates for p and E if Et and px are to be dimensionally identical .
Now let us consider an association between the variables$$(t,x,y,z,x_1,x_2,x_3.....)$$ and $$(E,p_x,p_y,p_z,p_1,p_2,p_3.....)$$. Here $$x_1,x_2,x_3 etc$$ are some fundamental quantities which are "hidden" or "curled up " in nature---variables that cannot be accessed by presently available experimental methods.
In place of (1) we now have,
$$K(E,p)=fg=ce^{-ia\Sigma(Et-p_1x_1-p_2x_2-p_3x_3-p_jx_j)}$$
We may think of an extended Lorentz transformation for an arbitrary boost in the $$x_i-{x_i}'$$ direction having exactly the sane form of our known Lorentz transformations.
$$x'_i=\gamma(x_i-vt)$$
$$t'=\gamma(t-v/c^2 x_i)$$
Other components will not change.
v is the n dimensional boost having a non-zero component only in the [latex]x_i-x'_i[[/latex] direction."c " is the speed of "n-dimensional " light. Incidentally we are working in an n+1 dimensional system (with the inclusion of time)
If xi and x'i do not correspond to our known spatial coordinates x , y or z then two observers spatially at rest(relative motion being zero) will have different different clock rates if they have a relative motion in the xi-x'i direction.This corresponds to a situation in GR where two observers at rest at different potentials have different clock rates(example:GPS). Difference in clock rates at different points is a typical feature of curved space.
Now our 3D "c" corresponds to light as we perceive in the known world. It matches with the concept of "photons"
Can the n-Dimensional "c" match with something called the "Graviton".
Can the n-Dimensional "c" lead to other type of "on s" that is to "Other-ons"?
[In the multidimensional case there are certain modes(or oscillation modes) in the "hidden dimensions" that causes gravity according to our model. You may consider the difference in the clock rates as an indication]
NB: The transformation given in the first line becomes a Fourier Transformation when both x and t tend to infinity
$$F(E,p,x,t)=\int_{-\infty}^{x}\int_{-\infty}^{t}f(x,t)e^{ia(Et-px)}dxdt$$
$$\frac{\partial F}{\partial x}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial x}e^{ia(Et-px)}-f(x,t)iap e^{ia(Et-px)}]dxdt$$
$$\frac{\partial F}{\partial x}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial x}-f(x,t)iap )e^{ia(Et-px)}]dxdt$$----(1)
Similarly,
$$\frac{\partial F}{\partial t}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial t}e^{ia(Et-px)}+f(x,t)iaE e^{ia(Et-px)}]dxdt$$
Or,
$$\frac{\partial F}{\partial t}=\int_{-\infty}^{x}\int_{-\infty}^{t}[\frac{\partial f}{\partial t}+f(x,t)iaE )e^{ia(Et-px)}]dxdt$$----(2)
If $$\frac{\partial F}{\partial x}=0$$ we have from (1) as an option:
$$\frac{\partial f}{\partial x}-f(x,t)iap =0$$
or,$$f(x,t)=Ae^{iapx}$$---- (3)
Again if $$\frac{\partial F}{\partial t}=0$$
then,
$$\frac{\partial f}{\partial t}+f(x,t)iaE=0$$
Or,$$f(x,t)=Be^{-iaEt}$$ ------------- (4)
We may write,
$$f(x,t)=Ce^{-ia(Et-px)}$$ -------------- (5)
And proceed as in the previous manner.
Inserting (5) into(1) we have:
$$F(E,p)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}Ce^{-ia(Et-px)}e^{ia(E_0t-p_0x)}dxdt$$
Or,
$$F(E,p)=Const \times \del (E-E_0)\del (p-p_0)$$ ------------ (6)
We now consider,
$$\psi (x,t)=\Sigma C_ie^{-ia(E_it-p_ix)}$$ ----------- (7)
By inserting (7) into (1) we have,
$$F(E,p)= \Sigma C_i\del (E_i-E_0)\del (p_i-p_0)$$ --------(8)
F(E,p) is the "Transformation " of $$\psi (x,t)$$, the transformation being defined on the first line of the post.It becomes athe Fourier Transform if x and t are allowed to tend to infinity.
F(E,p) if integrated over the (E,p) domain , it counts the number of modes having $$E=E_0$$ and $$p=p_0$$
If the number of such nodes corresponding to some $$(E_0,p_0)$$ is divided by the total number of possible modes we get a probability picture in an expected manner.By Fourier inversion we should get the same picture on the (x,t) domain.
The periodic nature of psi is clear from the expression (7) or (5).Psi is indeed a "wave function" By using suitable boundary conditions we restrict ourselves to discrete values of E and p.
Observations:
1. Equations (5) and (7) satisfy the Klein Gordon Equations.
2 Invariance of the exponent in (5) or (7) may be achieved through the Lorentz transformations provide "a" is an invariant. This corresponds to the fact that $$\hbar$$ is an universal constant.
3.If x and t represent represent x-coordinate(spatial) and time respectively then momentum and energy would be suitable candidates for p and E if Et and px are to be dimensionally identical .
Now let us consider an association between the variables$$(t,x,y,z,x_1,x_2,x_3.....)$$ and $$(E,p_x,p_y,p_z,p_1,p_2,p_3.....)$$. Here $$x_1,x_2,x_3 etc$$ are some fundamental quantities which are "hidden" or "curled up " in nature---variables that cannot be accessed by presently available experimental methods.
In place of (1) we now have,
$$K(E,p)=fg=ce^{-ia\Sigma(Et-p_1x_1-p_2x_2-p_3x_3-p_jx_j)}$$
We may think of an extended Lorentz transformation for an arbitrary boost in the $$x_i-{x_i}'$$ direction having exactly the sane form of our known Lorentz transformations.
$$x'_i=\gamma(x_i-vt)$$
$$t'=\gamma(t-v/c^2 x_i)$$
Other components will not change.
v is the n dimensional boost having a non-zero component only in the [latex]x_i-x'_i[[/latex] direction."c " is the speed of "n-dimensional " light. Incidentally we are working in an n+1 dimensional system (with the inclusion of time)
If xi and x'i do not correspond to our known spatial coordinates x , y or z then two observers spatially at rest(relative motion being zero) will have different different clock rates if they have a relative motion in the xi-x'i direction.This corresponds to a situation in GR where two observers at rest at different potentials have different clock rates(example:GPS). Difference in clock rates at different points is a typical feature of curved space.
Now our 3D "c" corresponds to light as we perceive in the known world. It matches with the concept of "photons"
Can the n-Dimensional "c" match with something called the "Graviton".
Can the n-Dimensional "c" lead to other type of "on s" that is to "Other-ons"?
[In the multidimensional case there are certain modes(or oscillation modes) in the "hidden dimensions" that causes gravity according to our model. You may consider the difference in the clock rates as an indication]
NB: The transformation given in the first line becomes a Fourier Transformation when both x and t tend to infinity
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