twistor59 wrote:

Teuton wrote:
That is to say, I generally deny that there can be 2-, 1-, or 0-dimensional things in physical reality.

(my bold)
Quite a strong statement (the bit in bold). Do you mean that it's something that you intuitively feel to be the case or is it something that you can prove (at least semi) rigorously ?

Yes, it's quite a strong statement based on the conceptual intuition that points, lines, and surfaces are merely mathematically abstracted and idealized spatial boundaries of physical objects (substances). They are objects of physical thought but not independently existing physical objects.

Teuton wrote:

twistor59 wrote:

Teuton wrote:
That is to say, I generally deny that there can be 2-, 1-, or 0-dimensional things in physical reality.

(my bold)
Quite a strong statement (the bit in bold). Do you mean that it's something that you intuitively feel to be the case or is it something that you can prove (at least semi) rigorously ?

Yes, it's quite a strong statement based on the conceptual intuition that points, lines, and surfaces are merely mathematically abstracted and idealized spatial boundaries of physical objects (substances). They are objects of physical thought but not independently existing physical objects.

So what exists 'independently' in your world view

Seems like the old metaphysical conundrum of what sort of reality abstract ideas have.

twistor59 wrote:A quick google around reveals some papers which purport to be about the 2D hydrogen atom, like this one. However they treat the Coulomb potential as the 3d function (i.e. 1/r) just restricted to 2 dimensions instead of using the "correct" 2D potential value, which is log(r).

Turns out to have a solution similar to the full-3D solution.

On a related note, I had a go a while back at computing the Klein Gordon propagator in various dimensions.

In effect, we must solve Dx2P - d2P/dt2 - m2P = delta(x)*delta(t)
Our first stop is
P(x,t) = integrate over all pn,E : (1/(2pi)n+1)) * (exp(i*p.x-i*E*t)/(E2 - p2 - m2 + i*eps)

I found a paper on doing it in 4D, and we can generalize from there. Equation 10:
P(x,t) = integrate over all pn : -(i/2) * (1/(2pi)n)) * (exp(i*p.x-i*E*t)/E)
where E = sqrt(p2 + m2)

Integrating over all time gives a pure-space version:
P(x) = integrate over all pn : - (1/(2pi)n)) * (exp(i*p.x)/E2)

One can do these integrals with the help of Abramowitz and Stegun: Handbook of Mathematical Functions -- yes, all the formulas are scanned into there, though not much of the tables. I also have the help of Mathematica -- computer-algebra software. If you are going to do a lot of theoretical work, I recommend getting some computer-algebra software.

Integrating over all angles we can do by turning the P(x,t) and P(x) integrals into
integrate p: 0 to +oo, a: 0 to pi : (1/(2pi)n) * pn-1 * (sin(a))n-2 * Wn-1 * exp(i*p*r*cos(a)) *
{ - (i/2)*exp(-i*E/t)/E or - 1/E2}
(A&S 9.1.20, p.360) where the n-D total solid angle Wn = 2*pin/2/Gamma(n/2) -- Gamma(n+1) = n!.

One gets for P(x,t) and P(x):
integrate p: 0 to +oo : (1/(2pi)n/2) * Jn/2-1(p*r)/(p*r)n/2-1 pn-1 * { - (i/2)*exp(-i*E/t)/E or - 1/E2}

where J is a Bessel function. The P(x) one is relatively easy, at least with "Hankel-Nicholson integrals" (A&S 14.4.44, p.488):

P(x) = - (1/(2pi)n/2 (m/r)n/2-1 * Kn/2-1(m*r)

For n >= 3, the m -> 0 limit is
- (1/pin/2) * Gamma(n/2-1) * r-(n-2) = - 1/(Wn*(n-2)*rn-2)
(A&S 9.6.9, p.375) -- which one can find more directly.

For n = 2, that limit is + (1/(2pi)) * (log(m*r/2) + (Euler-Mascheroni constant))

For n = 1, that limit is - (1/2)*(1/m - r)

We get some simple forms for some of the solutions:
For n = 1, - (1/2) * (1/m) * exp(-m*r)
For n = 3, - (1/(4*pi)) * (1/r) * exp(-m*r)

With the help of (A&S 9.1.31 p.361, 9.6.28 p.376), different dimensions are interrelated by
Pn+2(r) = - (1/(2pi)) * (1/r)(d/dr)Pn(r)

something also true of P(r,t). This means that we only need to solve the n = 1 and n = 2 cases of P(r,t); the rest is by
Pn+2(r,t) = - (1/(2pi)) * (1/r)(d/dr)Pn(r,t)

I'll save those n = 1 and n = 2 cases for my next post.

lpetrich wrote:

twistor59 wrote:A quick google around reveals some papers which purport to be about the 2D hydrogen atom, like this one. However they treat the Coulomb potential as the 3d function (i.e. 1/r) just restricted to 2 dimensions instead of using the "correct" 2D potential value, which is log(r).

Turns out to have a solution similar to the full-3D solution.

On a related note, I had a go a while back at computing the Klein Gordon propagator in various dimensions.

In effect, we must solve Dx2P - d2P/dt2 - m2P = delta(x)*delta(t)
Our first stop is
P(x,t) = integrate over all pn,E : (1/(2pi)n+1)) * (exp(i*p.x-i*E*t)/(E2 - p2 - m2 + i*eps)

I found a paper on doing it in 4D, and we can generalize from there. Equation 10:
P(x,t) = integrate over all pn : -(i/2) * (1/(2pi)n)) * (exp(i*p.x-i*E*t)/E)
where E = sqrt(p2 + m2)

Integrating over all time gives a pure-space version:
P(x) = integrate over all pn : - (1/(2pi)n)) * (exp(i*p.x)/E2)

One can do these integrals with the help of Abramowitz and Stegun: Handbook of Mathematical Functions -- yes, all the formulas are scanned into there, though not much of the tables. I also have the help of Mathematica -- computer-algebra software. If you are going to do a lot of theoretical work, I recommend getting some computer-algebra software.

Integrating over all angles we can do by turning the P(x,t) and P(x) integrals into
integrate p: 0 to +oo, a: 0 to pi : (1/(2pi)n) * pn-1 * (sin(a))n-2 * Wn-1 * exp(i*p*r*cos(a)) *
{ - (i/2)*exp(-i*E/t)/E or - 1/E2}
(A&S 9.1.20, p.360) where the n-D total solid angle Wn = 2*pin/2/Gamma(n/2) -- Gamma(n+1) = n!.

One gets for P(x,t) and P(x):
integrate p: 0 to +oo : (1/(2pi)n/2) * Jn/2-1(p*r)/(p*r)n/2-1 pn-1 * { - (i/2)*exp(-i*E/t)/E or - 1/E2}

where J is a Bessel function. The P(x) one is relatively easy, at least with "Hankel-Nicholson integrals" (A&S 14.4.44, p.488):

P(x) = - (1/(2pi)n/2 (m/r)n/2-1 * Kn/2-1(m*r)

For n >= 3, the m -> 0 limit is
- (1/pin/2) * Gamma(n/2-1) * r-(n-2) = - 1/(Wn*(n-2)*rn-2)
(A&S 9.6.9, p.375) -- which one can find more directly.

For n = 2, that limit is + (1/(2pi)) * (log(m*r/2) + (Euler-Mascheroni constant))

For n = 1, that limit is - (1/2)*(1/m - r)

We get some simple forms for some of the solutions:
For n = 1, - (1/2) * (1/m) * exp(-m*r)
For n = 3, - (1/(4*pi)) * (1/r) * exp(-m*r)

With the help of (A&S 9.1.31 p.361, 9.6.28 p.376), different dimensions are interrelated by
Pn+2(r) = - (1/(2pi)) * (1/r)(d/dr)Pn(r)

something also true of P(r,t). This means that we only need to solve the n = 1 and n = 2 cases of P(r,t); the rest is by
Pn+2(r,t) = - (1/(2pi)) * (1/r)(d/dr)Pn(r,t)

I'll save those n = 1 and n = 2 cases for my next post.

OK I'll have a look at my notes tonight to see if I got the same... !

Good idea.

Returning to the original form of P(x,t), I notice that the exponential function contains the expression
p.x - E*t
Taking the projection of p along the direction of x, px, this becomes
px*x - E*t

We can now do a Lorentz boost on px and E to put that expression in some more convenient form.

For a spacelike interval, |x| > |t|, and we take
R = sqrt(t2 - x2)
p -> (p*x + E*t)/sqrt(x2 - t2)
E -> (p*t + E*x)/sqrt(x2 - t2)
giving
p*x - E*t -> p*sqrt(x2 - t2)

For a timelike interval, |t| > |x|, and we take
T = sgn(t)*sqrt(t2 - x2)
p -> (p*t + E*x)/T
E -> (p*x + E*t)/T
giving
p*x - E*t -> E*T
where sgn(t) = 1 for t > 0 and -1 for t < 0
Note that a Lorentz boost preserves the direction of time.

In both cases, E2 - p2 is unchanged. The integrals become much easier.
P(x,t) = integrate over all pn: - (i/2)(1/(2pi)n) * (1/E) * {exp(i*px*R), exp(-i*E*T)}

This result is for eps small and positive. If eps is small and negative, then one gets its complex conjugate. Since we want a real-valued propagator, we take the real part, the Cauchy principal value.

In the spacelike case, the integrand is an even function of px, which is integrated from -oo to +oo. That makes the integral imaginary, and therefore not contributing. Thus, P(x,t) = 0 for spacelike (x,t).

In the timelike case, however, the integrand has a nonvanishing real part:

P(x,t) = integrate over all pn: - (1/2)(1/(2pi)n) * (sin(E*T)/E)

This integral becomes

P(x,t) = integrate over p: 0, +oo: - (1/2)(Wn/(2pi)n) * pn-1 * (sin(E*T)/E)

For the first two numbers of dimensions,

P1(x,t) = - (1/2) * sgn(T) * (1/2) * J0(m*T)
P2(x,t) = - (1/2) * (1/(2pi)) * cos(m*T)/T = - (1/2) * sgn(T) * (1/(2pi)1/2)) * (m/|T|)1/2 * Y1/2(m*|T|)
Recurrence:
Pn+2(x,t) = 1/(2pi) * (1/T)(d/dT) Pn(x,t)

P2k+1 = - (1/2) * sgn(T) * (1/(2pi)k) * (-1)k * (m/T)k * Jk(m*T)
P2k+2 = - (1/2) * sgn(T) * (1/(2pi)k+1/2) * (-1)k * (m/|T|)k+1/2 * Yk+1/2(m*|T|)

(A&S, the Bessel-function chapters)

These propagators are (retarded) - (advanced). To make them pure retarded, turn sgn(T) into 2*H(T) , where H(T) is the Heaviside step function: 0 for T < 0, 1 for T > 0.

Let's now take the massless limit.

P1(x,t) = - (1/2) * sgn(T) * (1/2)
P2(x,t) = - (1/2) * (1/(2pi)) * (1/T)

P2k+1(x,t) = 0
P2k+2(x,t) = - (1/2) * (1/(2pi)k) * (2k-1)!! / T2k+1

Leaving out what happens on the light cone; one has to take derivatives of H(t2 - r2) there. In particular, here is the complete expression for the first three numbers of space dimensions:
P1(x,t) = - (1/2) * sgn(T) * (1/2) * H(|t| - r)
P2(x,t) = - (1/2) * sgn(T) * (1/(2pi)) * (H(|t| - r)/|T|)
P3(x,t) = - (1/2) * sgn(T) * (1/(4pi*r)) * delta(r - |t|)
with similar expressions for more dimensions.

how is the Klein-Gordon equation relevant here?

Mononoke wrote:how is the Klein-Gordon equation relevant here?

Oh it's just that it's the prototype for many QFT calculations. The fancier ones (Maxwell etc) just have a few extra indices relating to polarization etc, but they behave similarly to the KG case as far as the propagator is concerned.

twistor59 wrote:

OK I'll have a look at my notes tonight to see if I got the same... !

I looked up what it was I was doing (it was a year or so ago). Basically it was an exercise from Tony Zee's book, deriving the force law in various dimensions via the propagator.

The arguments went roughly like this:

We have the functional integral

Z = ∫Dφ exp(i∫d4x[1/2φ(𝜕2+m2)φ+Jφ])


which is essentially (formally) solved by

exp(-i/2 J.A-1.J)

where A-1 is the inverse of the differential operator (𝜕2+m2), in other words the propagator, i.e. solution, D, of

-(𝜕2+m2)φ = 𝛿4(x-y)

Writing Z(J) = exp iW(J) and treating W as an action, the energy (since action is energy x time) becomes, after putting a couple of delta functions in for the sources:

E = -∫d3k exp(ik.(x1 - x2 )) / (k2+m2)


Explicitly doing this integral gives us the force law that we're after. Zee does the three dimensional case in the book and gets

E = -(1/r)exp(-mr)

(I can't be bothered writing the constant factors).

I had a go at the 2 space dimension case as follows:

rewriting things in polar coordinates and expanding the exponential with trig functions, I ended up with

∬ cos(krcos𝜃)/(k2 + m2) d𝜃kdk

(k integral from 0 to ∞; 𝜃 integral from 0 to π )

Giving

π ∬J0(kr)/(k2+m2)kdk

(k integral from 0 to ∞ )

This is the Hankel transform of 1/(k2+m2), which has approx behaviour

-ln(mr)

This agrees pretty much with your result except you have an extra constant in there.

For the higher dimensional cases I proved that you end up with essentially

powers of (1/r)𝜕/𝜕r times the 2 or 3 dimensional cases, depending on whether we're looking at odd or even dimensions

Its been proposed that life could exist on Enceladus (one of Saturns moons), under the surface in water melted by the gravitational influence of Saturn itself. If that turns out to be true, I wonder if you friend will claim that god put that moon in the right orbit around Saturn in order to support that life?

twistor59 was interested in seeing what it looked like in different dimensions, and I'm also interested in that sort of question, so I calculated it.

The Klein-Gordon propagator is the solution of the scalar, spin-0 relativistic wave equation with an instant point source. One gets close relatives of this function for fields with nonzero spin, so solving the KG equation is a good starting point.

As one might expect, it is zero for all space-time points with a spacelike separation from the source point. This means that it has well-defined time ordering and causality, with nonzero values only inside the source point's light cones (timelike-interval regions).

The propagator has a big jump at the light-cone boundary, meaning that there's an advancing front that travels at c. What's behind it depends on the field's mass and the number of space dimensions. In the massless case and an even number of space dimensions, there's a tail behind the advancing front that extends to the source's spatial position. In the massless case and an odd number of spatial dimensions, the field is constant for 1 space dimension and confined to the advancing front for higher numbers of dimensions. In fact, for 3D, the advancing front duplicates the source's time behavior. Higher odd numbers of dimensions include derivatives of that source behavior. In the massive case, the mass makes an oscillating tail behind the advancing front.

^^ I see. That's where all this stuff gets tied into a thread about dimensions