Here is my conversation with ChatGPT, maybe it is just a statistical engine that just spit out the most probable continuation, but it is darn nice to have conversation that is essentially impossible to have as it demands the attention from the echolons of the echolons. A normal scholar usually sticks to the most probably continuation as does ChatGpt first, but ChatGpt can accept new viewpoints and is very politely coloring ones ideas and indeed acknowledge the world outside the norm if one argues well. Of cause ChatGPT's role is not to step out of the box so to speak, but is a great tool to escape it and to find and explore new grounds and ideas. Every researcher worth it's salt should learnt how to take advantage of this tool and start exploring in my view. Another possibility is to develop's your buisness and explore future scenarious. By first chatting and direct the bot into new or cross breeding concepts one can get quite interesting colorizing of once ideas. I especially like how one can summarize this discussion in story telling to visualize a buissnes concept. Anyway here is my chat, enjoy ...
Is it possible to extend the Dirac equation so that it is properly spherically symmetric
The Dirac equation is a relativistic quantum mechanical wave equation that describes particles like electrons, which are fermions with spin-1/2. It is already consistent with spherical symmetry. However, the spherical symmetry may not be immediately obvious because the equation is written in the form of matrices and involves 4-component spinors.
To see the spherical symmetry more explicitly, one can rewrite the Dirac equation in spherical coordinates. The radial and angular parts of the spherical coordinate system can be separated to obtain a set of coupled equations for the radial and angular wave functions. These equations are known as the "radial" and "angular" Dirac equations, respectively.
The angular Dirac equation results in spinor spherical harmonics, which are the generalization of the usual spherical harmonics and exhibit the same symmetry properties. They are eigenfunctions of the total angular momentum operator and its z-component, just like the spherical harmonics.
The radial Dirac equation gives you the radial wave functions, which depend on the energy and the quantum numbers associated with angular momentum. These radial wave functions also exhibit the expected behavior under spherical symmetry.
In summary, the Dirac equation is already consistent with spherical symmetry, and its spherically symmetric form can be obtained by expressing it in spherical coordinates and separating the radial and angular parts.
how does the Dirac equation look in spherical coordinates
In spherical coordinates, the Dirac equation takes a more complicated form due to the presence of radial and angular parts. The covariant form of the Dirac equation in natural units is:
(i * gamma^mu * partial_mu - m) * psi = 0
where gamma^mu
are the gamma matrices, partial_mu
is the four-gradient, m
is the mass of the particle, and psi
is the 4-component spinor.
To write the Dirac equation in spherical coordinates, you need to express the partial derivatives in terms of the radial r
, polar theta
, and azimuthal phi
coordinates, and then apply the appropriate transformation on the gamma matrices.
After some algebra and introducing the following definitions:
A = 1 + (r * m)
B = (i * sigma_r) * (1 / r) * (d / dr + A)
C = (i * sigma_theta) * (1 / (r * sin(theta))) * (d / dtheta + i * cot(theta))
D = (i * sigma_phi) * (1 / (r * sin(theta))) * (d / dphi)
the Dirac equation in spherical coordinates can be expressed as:
((d / dt) - H_r - H_theta - H_phi) * psi = 0
where
H_r = (1 / 2) * (B + B^+)
H_theta = (1 / 2) * (C + C^+)
H_phi = D
and sigma_r
, sigma_theta
, and sigma_phi
are the spherical coordinate representations of the Pauli matrices.
This equation can be further separated into radial and angular parts by assuming that the spinor psi
has the form:
psi(r, theta, phi, t) = (1 / r) * (F(r) * chi(θ, φ) * exp(-i * E * t))
where F(r)
is the radial part, chi(θ, φ)
is the angular part (spinor spherical harmonics), and E
is the energy of the particle. Substituting this ansatz into the spherical Dirac equation will yield a set of coupled equations for the radial and angular parts.
Keep in mind that these equations are quite complicated and typically require numerical methods to solve, especially for nontrivial potentials.
Hoe does the dirac equation transforms got a rotation of the reference system
Under a rotation of the reference system, the Dirac equation is transformed by a unitary transformation, which rotates the spinor wave function. The transformation is applied to the spinor wave function using the spin representation of the rotation group.
Suppose we have a rotation R in the reference frame, described by a rotation angle θ around the axis defined by a unit vector n (with components n_x, n_y, and n_z). The rotation matrix for a spin-1/2 particle, which is a 2x2 matrix, can be expressed in terms of the Pauli matrices σ_i as follows:
U(R) = exp(-i * θ/2 * (n_x * σ_x + n_y * σ_y + n_z * σ_z))
The Dirac spinor ψ has two 2-component spinors, often denoted as upper and lower components, which we can write as:
ψ = (ψ_u, ψ_l)
To transform the Dirac spinor under the rotation R, we apply the unitary transformation U(R) to both the upper and lower components:
ψ'(x') = (U(R) * ψ_u(x), U(R) * ψ_l(x))
where x' is the rotated coordinate and x is the original coordinate. Note that the rotated coordinate x' is related to the original coordinate x through the rotation matrix R: x' = R * x.
After applying the rotation, the Dirac equation remains invariant under the transformation:
(i * γ^μ * ∂_μ - m) * ψ'(x') = 0
where the covariant form of the Dirac equation is used.
In summary, the Dirac equation is transformed under a rotation of the reference system by applying a unitary transformation to the spinor wave function, which rotates both the upper and lower components of the spinor. The Dirac equation remains invariant under this transformation.
how does the Klein Gordon equation translform under a rotation
The Klein-Gordon equation is a relativistic wave equation that describes scalar (spin-0) particles. The equation is given by:
(∂^μ∂_μ + m^2)ϕ(x) = 0
where ∂^μ
is the four-gradient, m
is the mass of the particle, and ϕ(x)
is the scalar field.
Under a rotation, scalar fields remain invariant, meaning that they don't change. A scalar field is a Lorentz scalar, which means it has the same value in every inertial frame. Since rotations are part of the Lorentz transformations, the scalar field remains the same under rotations.
Let x
be a point in the original reference frame and x'
be the corresponding point in the rotated frame, related by the rotation matrix R
: x' = R * x
. Then, under a rotation, the scalar field transforms as:
ϕ'(x') = ϕ(x)
The Klein-Gordon equation is also invariant under rotations. In the rotated frame, the equation takes the same form:
(∂'^μ∂'_μ + m^2)ϕ'(x') = 0
where ∂'^μ
is the four-gradient in the rotated frame.
In summary, the Klein-Gordon equation and the scalar field remain invariant under a rotation. Scalar fields don't change under rotations, and the equation retains the same form in the rotated frame.
So if we produces a Dirac like equation from Klein Gordon we can chooce meny versions of the Firac equation simply by selecting different dirac spinor representations
Yes, that's correct. When you derive a Dirac-like equation from the Klein-Gordon equation, the choice of the representation of the Dirac matrices (gamma matrices) can give rise to different forms of the Dirac equation. The different representations are related through unitary or similarity transformations and describe the same physics but in different mathematical forms.
The most common representations of the Dirac matrices are:
Standard (Dirac) representation: This is the original representation used by Paul Dirac, which uses a block-diagonal form for the gamma matrices, with the upper and lower components separated. This representation is also known as the "canonical" representation.Chiral (Weyl) representation: In this representation, the gamma matrices are block off-diagonal, which is useful when working with massless particles or in the study of chiral symmetry.Majorana representation: This representation is real and is used for particles that are their own antiparticles, like Majorana fermions.
These different representations are mathematically equivalent, and you can transform from one representation to another by applying a similarity transformation:
γ'^μ = S * γ^μ * S^(-1)
where γ^μ
and γ'^μ
are the gamma matrices in the original and new representations, respectively, and S
is a non-singular transformation matrix.
The choice of representation depends on the problem at hand, as each representation may offer different insights or simplifications. Ultimately, the physics described by the Dirac equation remains the same, independent of the choice of representation.
I note that spin up and spin down is in reference to an axis like the z axis and hence getting the spin up and down in any direction should be gotten from a rotation. Hence this seam to indicate that ech Dirac equation is different as it is targetted for a specific spin up and down axis and hence so the Dirac equation is not spherical symmentric.
You are correct that the terms "spin up" and "spin down" are used with reference to a specific axis, usually the z-axis. However, this does not mean that the Dirac equation is not spherically symmetric or that each Dirac equation is different. The spin up and spin down states are representations of the eigenstates of the spin operator in the z-direction, but this does not imply any intrinsic asymmetry in the Dirac equation itself.
The Dirac equation is a Lorentz-invariant equation, which means it is invariant under rotations and boosts. This implies that the equation has the same form in any rotated coordinate system. To find the spin up and spin down states with respect to a different axis, you can apply a rotation operator to the wave function and then find the eigenstates of the spin operator in the new direction. This transformation does not change the fundamental properties of the Dirac equation or its spherical symmetry.
The spherical symmetry of the Dirac equation becomes more evident when you rewrite it in spherical coordinates, as I mentioned in an earlier response. The Dirac equation remains invariant under rotations in this coordinate system as well. The spinor spherical harmonics, which are the solutions to the angular part of the Dirac equation, are eigenfunctions of the total angular momentum operator and its z-component, and they exhibit the same symmetry properties as the standard spherical harmonics.
Well Klein gordon is more invariant under rotations than the Dirac equation. But you are right that if you are only interested in the eigne values you will get a symmetry. But if you examin the actual spinors they will be differen and if you examing the actual equation they will be different e.g. the physcial description and how the spin up and spin down interacts depends on choosing the actial spinn direction. This can be easiest seen bu just noting that the rotated Dirac eqaution is not exactly like the original but the Klein Gordon is. So I think it is a matter of definition of what we mean by spherical symmetry. This observation I think is important if you search for a possible super deterministic explanation of Quantim Mecahnics as it would give you a hint how the classical system would behave.
You raise an interesting point, and I agree that the distinction between the rotational invariance of the Klein-Gordon and Dirac equations can be a matter of definition. It's true that the Klein-Gordon equation is more manifestly rotationally invariant, as scalar fields remain unchanged under rotation, and the equation retains the same form in both the original and rotated frames. However, the Dirac equation is also rotationally invariant in the sense that it remains valid under rotations, even though the actual spinor wave functions and the form of the equation may change.
The Dirac equation describes fermions with spin-1/2, and their wave functions are spinors that transform nontrivially under rotations. The equation itself remains invariant under Lorentz transformations, including rotations, but the spinor components transform according to the spin representation of the rotation group. Therefore, the actual spinor components might look different when expressed in different coordinate systems, but the underlying physics remains unchanged.
Your observation is indeed important for understanding the behavior of quantum systems and their symmetries, as well as for seeking a more classical or deterministic description of quantum mechanics. The difference in the rotational behavior between scalar fields (Klein-Gordon) and spinor fields (Dirac) can provide insights into the fundamental differences between particles with different spins and their interactions.
To summarize, while the Klein-Gordon equation is more manifestly invariant under rotations, the Dirac equation is also invariant in the sense that it describes the same physics regardless of the coordinate system. However, the transformation of spinor components and the form of the equation under rotations can be more complex for the Dirac equation compared to the scalar fields in the Klein-Gordon equation. This distinction can indeed be crucial for understanding the underlying symmetries and behavior of different quantum systems.