subtlety of the curl operator

[AntonioLao]

the spatial rotation of a vector quantity is describable by the curl. It is defined as the vector or outer or cross product of the differential vector operator [math]\nabla[/math], called del or nabla with any arbitrary vector. In 3D Cartesian notations, it is given by

[math]\nabla=\frac{\partial}{\partial x}\mathbf{i}+\frac{\partial}{\partial y}\mathbf{j}+\frac{\partial}{\partial z}\mathbf{k}[/math]

the nabla operator is a space vector rate of change differential operator in contradistinction to the time scalar rate of change differential operator.

What is the equivalent temporal rotation of a vector quantity? Can there exist a curl operator with respect to time? In almost all wave phenomena, this time rate of change operator seems to be functionally associated to the concept of frequency for all periodic functons. Therefore the subtlety lies between the physical concept of rotation and the concept of vibration or oscillation.

[AntonioLao]

more subtleties lie in the fact that almost all physical theories were based on the equivalence of the space operator and the time operator except Einstein's field equations of general relativity. For examples, see the following links of equations.

1. Klein-Gordon equation
An equation which hold for spinless particles. The time-dependent form is

where is the d'Alembertian, is the wavefunction, m is the mass of the particle, c is the speed of light, and is h-bar.

2. Schrodinger equation
The time-dependent one-dimensional Schrödinger equation is given by

where i is the imaginary unit, is the time-dependent wavefunction, is h-bar, V(x) is the potential, and is the Hamiltonian operator

3. wave equation

4. Maxwell's equations of radiation

http://scienceworld.wolfram.com/phys...Equations.html

5. Navier-Stokes equation of fluid mechanics

http://scienceworld.wolfram.com/phys...Equations.html

comparing them to Einstein's field equations at

http://scienceworld.wolfram.com/phys...Equations.html

[Guille]

if you give me what i, j and k stand for,

[math]\nabla=\frac{\partial}{\partial x}\mathbf{i}+\frac{\partial}{\partial y}\mathbf{j}+\frac{\partial}{\partial z}\mathbf{k}[/math]

I will give my answers.

[AntonioLao]

if you give me what i, j and k stand for

i, j, k are the unit vectors along the 3D Cartesian coordinate axes.

[Guille]

i, j, k are the unit vectors along the 3D Cartesian coordinate axes.

Then my answer is that for time, you substitude i,j,k by f,g,h which can be the unit vectors along the 3d quantum07ian coordinate axis.

[AntonioLao]

f,g,h which can be the unit vectors along the 3d quantum07ian coordinate axis.

For quantum systems, there are more than just three axes. At least, there are 3 spatial axes, 1 time axis in addition to the intrinsic axes such as isotopic spin of Yang-Mill field, spin axis of Dirac's field.

[Guille]

For quantum systems, there are more than just three axes. At least, there are 3 spatial axes, 1 time axis in addition to the intrinsic axes such as isotopic spin of Yang-Mill field, spin axis of Dirac's field.

When I wrote quantum it was follow by 07, I was refering to the member of this forum denominated "quantum07" and more spepecifically, to his article on time, and his prove of 3 dimensional time: not refering to quantum mechanics.Sorry for the confusion created.

quantum07,

if you read this, do you mind me calling the time dimensions in axis: f, g and h?

[AntonioLao]

his prove of 3 dimensional time:

have not read his article but my guess is that his proofs are mathematical not physical. Physically speaking, reality of our universe is based on the thermodynamic arrow of time as the constant increase of entropy.

[Guille]

have not read his article but my guess is that his proofs are mathematical not physical. Physically speaking, reality of our universe is based on the thermodynamic arrow of time as the constant increase of entropy.

Yes, it's mathematical-geometrical.

Can't entropy decreasy? isn't entropy a property of energy? if it is, how can it be that energy is conservd but entropy not?

[Guille]

What is the equivalent temporal rotation of a vector quantity? Can there exist a curl operator with respect to time?

can there anyway be temporal or time rotations? what is meant by rotations in this? is curl operator is about space, there can't be curl operator of time, I think there should be ____ ______ of time. And you would need first to define that. And define what you use to define that, etz...