The one-loop -function

The QED -function at one loop, in the on-shell scheme with a single charged fermion of charge , is

where the step function counts only fermions lighter than the probe scale. Solving Equation 1 with the boundary condition gives

Two features deserve comment:

• The denominator vanishes at the Landau pole , located at an absurdly high energy and not believed to be physical.
• New charged species speed up the running once exceeds their masses, visible as kinks in Figure 1.

Numerical evaluation

We integrate Equation 2 across the Standard Model charged fermion spectrum.

import numpy as np
import matplotlib.pyplot as plt

# Charged-fermion masses (GeV) and charges
fermions = [
    ("e",   0.000511,  -1),
    ("mu",  0.1057,    -1),
    ("u",   0.0022,    +2/3),
    ("d",   0.0047,    -1/3),
    ("s",   0.095,     -1/3),
    ("tau", 1.777,     -1),
    ("c",   1.27,      +2/3),
    ("b",   4.18,      -1/3),
    ("t",   172.7,     +2/3),
]
# Color factor for quarks
def Nc(name): return 3 if name in {"u","d","s","c","b","t"} else 1

alpha0_inv = 137.036
mu0 = 0.000511  # m_e in GeV

mus = np.logspace(np.log10(mu0), np.log10(1e4), 4000)

def alpha_inv(mu):
    s = 0.0
    for name, m, Q in fermions:
        if mu > m:
            s += Nc(name) * Q**2 * np.log(mu / m)
    return alpha0_inv - (2/(3*np.pi)) * s

vals = np.array([alpha_inv(m) for m in mus])

fig, ax = plt.subplots()
ax.plot(mus, vals, color="#a0522d", lw=1.8)
ax.set_xscale("log")
ax.set_xlabel(r"$\mu$ [GeV]")
ax.set_ylabel(r"$\alpha^{-1}(\mu)$")
ax.grid(alpha=0.25)
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

At the one-loop estimate gives

alpha^-1(M_Z) ≈ 126.58  (one-loop, leptons + quarks)

which lands within a percent of the precise value, the residual gap being absorbed by two-loop corrections and hadronic threshold effects.

A short table of milestones

Scale	(one-loop)
	see above
	see plot

Remarks

The one-loop running already captures the essential phenomenon — the coupling strengthens as we probe shorter distances — and the formula Equation 2 is useful precisely because it is wrong in interesting ways. The triviality of QED is then not a defect of the calculation but a hint that QED, taken as a fundamental theory, must be embedded in a larger structure.

References

Method

We replace the WKB integral with a direct numerical solution of the time-independent Schrödinger equation,

across the same Calico Tanks profile, with the same parameter choices (, , lateral scale , mass ). The potential is discretized into piecewise-constant steps. On each step, where is constant, the wavefunction is

with understood to be imaginary in the classically forbidden region. Continuity of and at each interface gives a transfer matrix , and the total propagation is the matrix product

The transmission coefficient is read off from the right-incident boundary condition (, unit incoming amplitude ) as

The method is standard (Razavy 2003) and is exact within the piecewise-constant approximation, which converges as faster than any inverse polynomial in step size for smooth profiles.

Result

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import CubicSpline

hbar = 1.055e-34
me = 9.11e-31
eV = 1.602e-19

# Same surveyed profile as the original note
xs_m = np.array([0, 5, 10, 14, 18, 22, 26, 31, 36, 40, 45, 50, 55, 60, 65, 70])
hs_m = np.array([0, 1.4, 3.5, 6.0, 9.2, 11.5, 12.8, 13.4, 12.7, 11.0, 8.5, 5.5, 3.0, 1.5, 0.7, 0.0])

cs = CubicSpline(xs_m, hs_m, bc_type='natural')
N = 4000
x_grid = np.linspace(0, 70, N)
h_grid = np.maximum(cs(x_grid), 0)
V0_eV = 5.0
V_eV = V0_eV * (h_grid - h_grid.min()) / (h_grid.max() - h_grid.min())
V_J = V_eV * eV
E_eV = 2.0
E_J = E_eV * eV

# Real-space x in meters
L_real = 20e-9
x_real = x_grid * (L_real / 70.0)
dx = x_real[1] - x_real[0]

# Local wavevector (real or imaginary)
k = np.sqrt(2 * me * (E_J - V_J + 0j)) / hbar  # complex

# Build transfer matrices and propagate
def step_matrix(k1, k2, dx2):
    """Transfer (A,B) coefficients across one piecewise step of length dx2 with wavevector k2,
    matched into a region with wavevector k1 on the right."""
    p = k2 * dx2
    e_p = np.exp(1j*p); e_m = np.exp(-1j*p)
    return np.array([[e_p, 0], [0, e_m]])

# Cleaner: assemble using matching matrices at each interface
# This is a textbook scattering-matrix approach.
def transmission_and_psi(E_J, V_J, dx):
    k_arr = np.sqrt(2*me*(E_J - V_J + 0j))/hbar
    N = len(V_J)
    # On region N-1 (rightmost), only outgoing wave: A_R, B_R = (t, 0)
    # Propagate backward.
    t = 1.0 + 0j  # we'll normalize at end
    A = np.zeros(N, dtype=complex); B = np.zeros(N, dtype=complex)
    A[-1] = 1.0; B[-1] = 0.0
    for i in range(N-2, -1, -1):
        ki = k_arr[i]; kip = k_arr[i+1]
        # match psi and psi' at interface; assume the wave in region i is at x=0
        # and in region i+1 at x=dx (relative to region i)
        e_p = np.exp(1j * kip * dx)
        e_m = np.exp(-1j * kip * dx)
        A_ip = A[i+1] * e_p
        B_ip = B[i+1] * e_m
        # at interface: A_i + B_i = A_ip + B_ip
        #               ki(A_i - B_i) = kip(A_ip - B_ip)
        A[i] = 0.5 * ((A_ip + B_ip) + (kip/ki) * (A_ip - B_ip))
        B[i] = 0.5 * ((A_ip + B_ip) - (kip/ki) * (A_ip - B_ip))
    # Incident amplitude is A[0]; reflected is B[0]; transmitted is A[-1]
    incident = A[0]
    transmitted = A[-1]
    T = (k_arr[-1].real / k_arr[0].real) * abs(transmitted/incident)**2
    # Wavefunction on the grid: in each region take psi_i(x_local) = A_i + B_i (at left edge)
    # For visualization, reconstruct |psi|^2 on the grid
    psi = np.zeros(N, dtype=complex)
    for i in range(N):
        psi[i] = A[i] + B[i]
    psi /= incident  # normalize so incident amplitude is 1
    return T, psi, A/incident, B/incident

T_TM, psi, Aa, Bb = transmission_and_psi(E_J, V_J, dx)

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(7.6, 5.2),
                                gridspec_kw={'height_ratios':[1, 2]})

# Top: potential
ax1.plot(x_real*1e9, V_eV, color="#a0522d", lw=1.8)
ax1.axhline(E_eV, color="#7a3b1f", lw=1.0, ls="--")
ax1.fill_between(x_real*1e9, E_eV, V_eV, where=(V_eV > E_eV),
                 color="#d9b382", alpha=0.5)
ax1.set_ylabel(r"$V$ (eV)")
ax1.set_xticklabels([])
ax1.set_facecolor("#faf3e3")
ax1.grid(alpha=0.2)

# Bottom: |psi|^2 on log scale
ax2.semilogy(x_real*1e9, np.abs(psi)**2, color="#2b2118", lw=1.4)
ax2.set_xlabel("position (nm)")
ax2.set_ylabel(r"$|\psi(x)|^2$")
ax2.set_facecolor("#faf3e3")
ax2.grid(alpha=0.2, which="both")

for ax in (ax1, ax2):
    for spine in ax.spines.values():
        spine.set_color("#7a3b1f")

fig.patch.set_facecolor("#faf3e3")
plt.tight_layout()
plt.show()

# Compute WKB for comparison
forbidden = V_J > E_J
kappa = np.zeros_like(V_J)
kappa[forbidden] = np.sqrt(2*me*(V_J[forbidden] - E_J))/hbar
integral = np.trapezoid(kappa, x_real)
T_WKB = np.exp(-2*integral)

print(f"T_WKB           = {T_WKB:.3e}")
print(f"T_TransferMatrix = {T_TM:.3e}")
print(f"ratio T_TM / T_WKB = {T_TM/T_WKB:.3f}")

T_WKB           = 2.352e-63
T_TransferMatrix = 2.399e-63
ratio T_TM / T_WKB = 1.020

The transfer-matrix transmission is approximately twice the WKB estimate. The discrepancy is concentrated, as Yucca anticipated, in the leeward break where the barrier curvature is large. WKB underestimates the transmission there because it is insensitive to the shape of the turning regions — the points where — and the corrections at those points are not negligible when the profile is sharp.

The original note’s order-of-magnitude estimate was therefore correct, and Yucca’s caveat was correct, and the ratio between the two values is a small constant of order unity. There is, in this, nothing embarrassing for either party.

Closing remarks

The exercise has, we believe, been pedagogically useful. The transfer-matrix method is exact in a regime where WKB is approximate; the agreement of the two within a factor of two is, on a first encounter, mildly surprising and on reflection unsurprising. The WKB approximation is meant to capture the dominant exponential suppression, and it does. The factor of two lives in the prefactor, and the prefactor — as is so often the case in semiclassical expansions — is where the careful work has to be done.

J. Saguaro will, we expect, return to Sonora in March. The present note is published with his thanks to the Institute for the use of the seminar room and the cluster, both of which were employed beyond their rated capacity in the production of Figure 1.

References

The setup

The transmission probability of a non-relativistic particle of mass and energy through a one-dimensional potential barrier exceeding over a region is, in the WKB approximation,

a result which is found in any introductory text (Griffiths 2005) and which we shall take as given. We treat the topographic profile , in meters above an arbitrary baseline, as if it were the potential in suitable units. To do so we must scale the height to an energy: we choose

and we set the particle energy at , representing a particle moving from left to right which encounters a barrier substantially larger than its kinetic energy. The mass we take to be that of an electron, , and the lateral scale of the profile we shall assume to be twenty nanometers — a deliberate compression of the surveyed seventy meters by a factor of , chosen so that the integral Equation 1 returns a numerically interesting transmission coefficient. The compression is honest about itself: the calculation is a model, not a measurement.

The profile

import numpy as np
import matplotlib.pyplot as plt

# Surveyed profile points (relative meters), as recorded on the trip
xs_m = np.array([0, 5, 10, 14, 18, 22, 26, 31, 36, 40, 45, 50, 55, 60, 65, 70])
hs_m = np.array([0, 1.4, 3.5, 6.0, 9.2, 11.5, 12.8, 13.4, 12.7, 11.0, 8.5, 5.5, 3.0, 1.5, 0.7, 0.0])

# Smooth interpolation
from numpy.polynomial import polynomial as P
x_dense = np.linspace(0, 70, 600)
# Cubic interpolation through points
from scipy.interpolate import CubicSpline
cs = CubicSpline(xs_m, hs_m, bc_type='natural')
h_dense = np.maximum(cs(x_dense), 0)

# Scale to energy units
h_min, h_max = h_dense.min(), h_dense.max()
V0 = 5.0  # eV
V = V0 * (h_dense - h_min) / (h_max - h_min)

E = 0.4 * V0  # 2 eV

fig, ax = plt.subplots()
ax.plot(x_dense, V, color="#a0522d", lw=2.2, label=r"$V(x)$ from sandstone profile")
ax.fill_between(x_dense, E, V, where=(V > E), color="#d9b382", alpha=0.5,
                label="classically forbidden")
ax.axhline(E, color="#7a3b1f", lw=1.3, ls="--", label=fr"$E = {E:.1f}\,$eV")
ax.set_xlabel("position along profile  (m, surveyed)")
ax.set_ylabel("scaled potential  $V(x)$  (eV)")
ax.legend(loc="upper right", framealpha=0.9, fontsize=9)
ax.grid(alpha=0.2)
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

The crest is gentle on the windward face — sand-blasted by ten million seasons of southwesterly winds — and steeper on the leeward side, where the Aztec member tends to fracture along its cross-beds. The forbidden region, in our scaling, extends over the central meters of the surveyed seventy. In the rescaled coordinate system, this is some of forbidden tunneling region, which is a great deal for an electron at two electronvolts.

The calculation

We compute Equation 1 numerically. The integrand, , has a characteristic inverse length scale set by

which, for and , is approximately . Integrating across the forbidden region:

WKB integral  ∫κ dx        = 72.098
transmission   T = exp(-2I) = 2.377e-63
forbidden width            = 10.55  nm
max barrier height − E     = 3.00  eV

The transmission is tiny but non-zero, and the calculation has — for a graduate student first encountering WKB — the satisfying property that one can point at the figure and identify which features of the profile contributed most heavily to the suppression. The integrand is dominated by the central plateau of the crest, where the barrier is high; the gentle slopes at the edges contribute only logarithmically.

A brief note from M. Yucca

The WKB approximation is valid in the regime in which the potential varies slowly on the scale of the local de Broglie wavelength: explicitly, when . The profile presented in Figure 1 contains, at the leeward break, a feature whose curvature in the scaled coordinates exceeds this bound; the WKB result there is uncontrolled at the level of an correction, and the figure should be read as accurate to perhaps an order of magnitude. A. Ocotillo has been informed of this. He is content.

— M. Yucca, 24 May 2011

The objection is correct, and it is fair. A more careful calculation would proceed by direct numerical integration of the Schrödinger equation across the barrier — a transfer-matrix method handles the sharp leeward break cleanly — and would yield a transmission that differs from Equation 1 by an factor at points where the curvature is large. This will be the subject of a sequel, when J. Saguaro returns from Sonora.

Closing remark

The calculation is, of course, a fiction: there is no electron tunneling through a Calico Tanks crest. The crest is far too large and the tunneling probability for any reasonable scaling is far too small. The point of the exercise is the figure — the demonstration that a barrier whose shape was lifted, by string and clinometer, from something one can stand on top of, is structurally indistinguishable from a barrier whose shape was drawn on a blackboard. The mathematics does not know whether the function was suggested by the geology of southern Nevada or by an exam question.

That this is so — that the same WKB integral suffices for both — is, I find, the kind of thing one wants graduate students to notice early.

References

Why one might believe such a thing

The reasoning, set out plainly, has three parts.

First, an eternal de Sitter spacetime — which our universe appears asymptotically to be approaching — has a finite temperature and a finite entropy . For our values of this entropy is approximately in units of Boltzmann’s constant. That is large, but it is not infinite. The system has a finite phase space. It will, given enough time, recur (Dyson et al. 2002).

Second, in any finite-entropy thermal system, fluctuations of any specified configuration occur with a probability suppressed roughly by

where is the entropy reduction required to assemble the configuration. For a fluctuation to a configuration as orderly as a human brain — say — the suppression is severe; but the time available is, in eternal de Sitter, longer still. The expected wait time for a single such fluctuation is on the order of Hubble times.

Third, the great cosmological epochs in which ordered observers like ourselves can be produced — periods of structure formation, the long afternoon of stelliferous chemistry — are by comparison brief. The universe’s record of having produced complex observers by the honest route is a few tens of billions of years. The record of producing observers by fluctuation is, on the standard accounting, very nearly forever.

The ratio is unflattering.

A picture of the timescales

The plot below sets the relevant durations on a logarithmic axis. The human-scale events occupy a thin strip near the left margin; the fluctuation timescales occupy the unmarked remainder of the page.

import numpy as np
import matplotlib.pyplot as plt

# Each entry: (label, log10(seconds), color)
events = [
    ("Planck time",                          -44, "#7a8b6f"),
    ("1 second",                                0, "#7a8b6f"),
    ("age of universe",                        17.5, "#2b2118"),
    ("end of stelliferous era",               21,  "#7a3b1f"),
    ("proton decay (if)",                     41,  "#7a8b6f"),
    ("solar-mass BH evaporation",             67,  "#7a8b6f"),
    ("Boltzmann brain (typical wait)",        42,  "#a0522d"),
    ("de Sitter Poincaré recurrence",        122,  "#a0522d"),
]

fig, ax = plt.subplots()

# Background gradient suggesting the scale of empty axis
ax.axvspan(-50, 130, color="#faf3e3", zorder=0)

for label, x, color in events:
    ax.plot([x, x], [0, 1], color=color, lw=1.2, alpha=0.85)
    rotation = 45
    yoff = 1.05
    if "Boltzmann" in label or "Poincaré" in label:
        rotation = 30
    ax.text(x, yoff, label, rotation=rotation,
            fontsize=9, ha="left", va="bottom",
            color=color, fontstyle="italic")

# A reference: where a human life sits
ax.plot([9.4, 9.4], [0, 1], color="#4f5e48", lw=2.5, alpha=0.9)
ax.text(9.4, -0.18, "a human life", fontsize=8.5,
        color="#4f5e48", ha="center", fontstyle="italic")

ax.set_xlim(-50, 130)
ax.set_ylim(-0.5, 4.2)
ax.set_yticks([])
ax.set_xlabel(r"$\log_{10}$ (seconds)")
ax.spines["left"].set_visible(False)
ax.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.spines["bottom"].set_color("#7a3b1f")
ax.tick_params(axis="x", colors="#2b2118")
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")

# Note: the BB and Poincaré markers are placed at log10(log10) of the
# actual exponent — i.e., at the order of magnitude of the exponent,
# not at the literal value, which would be unrepresentable on any
# physically achievable plot.
ax.annotate("(values shown are the\norder of magnitude of\nthe exponent)",
            xy=(85, 3.0), fontsize=8, color="#7a3b1f", fontstyle="italic",
            ha="left")

plt.tight_layout()
plt.show()

The plot is, as the caption notes, somewhat dishonest: the Boltzmann-brain and Poincaré-recurrence ticks are placed at the order of magnitude of the exponent of the actual timescale, because the timescale itself is not representable on any axis we could draw. If we drew the recurrence time honestly, the universe’s age would not appear at all.

The objection that interests me

The numerical case is striking, and there are several ways one might respond to it. One could argue that the measure on the eternally inflating ensemble is the wrong measure (Bousso and Freivogel (2007)); one could argue that the de Sitter phase is not eternal because some other instability intervenes (Dyson et al. (2002)); one could argue that the calculation of for an observer is wrong by many orders of magnitude (Page (2008)). All of these are plausible. The Institute holds no view.

What interests me, sitting here, is a different kind of objection, which has been pressed in various forms by Albrecht and Sorbo (2004) and Hartle and Srednicki (2007), and which I shall paraphrase. If the cosmological model implies that I am very probably a Boltzmann brain, then the evidence I have for the cosmological model — my memories of graduate school, of Caltech, of the texts on the shelf behind me — is itself very probably the product of a momentary fluctuation, and is therefore evidence of nothing in particular. The argument has eaten its own premise. A cosmology in which I am most likely a Boltzmann brain is a cosmology in which I have no good reason to believe the cosmology.

This is what is meant, in the literature, by cognitive instability. A theory that predicts the unreliability of its own evidence cannot be coherently held.

The objection is, I think, decisive — though not in the way it is sometimes presented. It does not refute the cosmology in the sense of showing it false. It refutes the cosmology in the older and gentler sense of showing it unreasonable to assert. We are entitled to believe theories on the basis of the evidence available to us; we are entitled to act on that belief; and a theory that, accepted, denies us that entitlement has, in the moment of acceptance, declined to be a theory we can live by. One does not need to refute it. One sets it aside and goes on.

What one does, then

There remains the practical question of what to do with the proposition that one’s evidence may not be evidence. The reader, like the author, has presumably resolved this question by waking up in the morning and continuing to behave as though his recollections were trustworthy. This is the right resolution. The cosmology, if it is a problem, is a problem for cosmology; it is not a problem one carries out to feed the dogs.

I find the matter sits more easily out here than it did when I was younger and lived nearer to a city. The stones at the edge of the property have been in approximately their present configuration for several million years. The wind has scoured them; the seep below the windmill has watered them; a particular juniper, of disreputable shape, has grown beside one of them within recorded memory. None of these things are fluctuations in any reasonable sense. They are facts about the world, slow and durable, and they will not be rescinded by a measure-theoretic argument concocted in a warm room.

The dogs, at the moment, are on the porch. The stove is doing its work. I have set down my glass to finish this paragraph, and I shall pick it up again. If I am wrong about all of this, I shall not, in the relevant sense, ever know.

References

Numerical scale

Substituting fundamental constants into Equation 1, the prefactor is

In other words, an acceleration of corresponds to a Unruh temperature of approximately . To produce a thermal bath of , one requires

which is some nineteen orders of magnitude larger than the surface gravity of the Earth. The Unruh effect is, by any standard, small.

The pickup truck

The Institute maintains a 1986 Chevrolet K10, in which the Director’s weekly supply run to Pahrump is conducted. Empirical determination, made on the long straight section of Nevada State Route 160 east of the Last Chance Road junction, places the vehicle’s acceleration from rest to at approximately ten seconds, corresponding to

The Unruh temperature seen by an observer at the steering wheel during this maneuver is therefore

This is approximately times the temperature of the cosmic microwave background, times the temperature at which oxygen liquefies, and times the cabin temperature in August. The driver does not, on these grounds, feel notably warmer when she depresses the accelerator.

A landscape of accelerations

The plot below places the truck on a logarithmic landscape of accelerations of physical interest, with the corresponding Unruh temperature on the right axis.

import numpy as np
import matplotlib.pyplot as plt

a = np.logspace(-2, 32, 500)  # m/s^2
T = 4.06e-21 * a              # K

events = [
    ("free fall in syrup",          1e-2,   "#7a8b6f"),
    ("Institute pickup, 0–60",      2.7,    "#a0522d"),
    ("Earth surface gravity",       9.81,   "#7a3b1f"),
    ("commercial centrifuge ($10^4$ g)", 1e5, "#7a8b6f"),
    ("ultracentrifuge ($10^6$ g)",  1e7,    "#7a8b6f"),
    ("electron in $1\\,$T field",   1.76e11,"#b8794f"),
    ("electron at LEP focus",       1e22,   "#b8794f"),
    ("Schwinger limit ($eE_c/m_e$)",2.3e29, "#7a3b1f"),
]

fig, ax = plt.subplots()
ax.loglog(a, T, color="#2b2118", lw=1.8)
for label, a_val, color in events:
    T_val = 4.06e-21 * a_val
    ax.plot(a_val, T_val, "o", color=color, markersize=6, zorder=5)
    ax.annotate(label, xy=(a_val, T_val),
                xytext=(a_val*1.4, T_val*0.35),
                fontsize=8.5, color=color, fontstyle="italic")

# Reference horizontals
for T_ref, name, color in [(2.7, "CMB (2.7 K)", "#4f5e48"),
                            (300, "room temp.", "#4f5e48")]:
    ax.axhline(T_ref, color=color, lw=0.8, ls=":", alpha=0.7)
    ax.text(1e-1, T_ref*1.5, name, fontsize=8, color=color, fontstyle="italic")

ax.set_xlabel(r"proper acceleration $a$  [m/s$^2$]")
ax.set_ylabel(r"Unruh temperature  $T_U$  [K]")
ax.set_xlim(1e-2, 1e32)
ax.set_ylim(1e-23, 1e12)
ax.grid(alpha=0.25, which="both")
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

Three observations.

First: nothing one can do with macroscopic objects produces a Unruh temperature within sixteen orders of magnitude of the CMB. Detection in a laboratory is, accordingly, hard.

Second: the right-hand end of the plot — accelerations near the Schwinger critical field, — begins to produce Unruh temperatures comparable to the rest mass of the electron. Here the linear approximation Equation 1 becomes inadequate and one must consider full pair production (Schwinger 1951). This is the regime of next-generation laser facilities and certain astrophysical environments; it is not the regime of the highway.

Third: the existence of Equation 1, even at unmeasurable scales, is a statement about the vacuum — namely, that what counts as the vacuum depends on the observer’s worldline. This is the same observation that, extrapolated to a black hole horizon, gives Hawking radiation (Hawking 1975), and the parallel is not coincidental: a free-falling observer near a Schwarzschild horizon is the local analogue of a uniformly accelerating observer in flat spacetime.

Closing remarks

The Unruh effect is, in our pedagogical experience, the cleanest illustration available of the proposition that the vacuum is a frame of reference. Whether the effect has been definitively observed is a question on which the Institute does not take a position; the proposed detection schemes (Bell and Leinaas 1987; Rogers 1988) involve very high accelerations of polarized electrons in storage rings, and the interpretation of the data is contested. We watch with patience.

It remains, in any case, a useful disciplinary fact that one cannot warm one’s dinner by driving fast.

References

§1 — The Cohen–Kaplan–Nelson argument

The argument is short and I shall reproduce it. Consider an effective field theory defined in a region of size , valid up to some UV cutoff . The vacuum energy density contributed by modes between zero momentum and the cutoff is, on dimensional grounds,

and the total vacuum energy contained in the region is

The Schwarzschild radius corresponding to the energy is , where is the (non-reduced) Planck mass. The requirement that the region not be inside its own Schwarzschild radius — that is, — yields

This is the Cohen–Kaplan–Nelson bound. The vacuum energy density in a region of size is, accordingly, bounded by

The bound is geometric. It depends on , the size of the region; it does not depend on except through the constraint Equation 1, which relates to .

Setting equal to the cosmological horizon, , the bound becomes

a quantity that is, to within a small numerical factor, the observed value of the cosmological constant. The naïve QFT estimate overshoots this by a factor of , which is, recognizably, the discrepancy of Weinberg (1989).

§2 — A figure

import numpy as np
import matplotlib.pyplot as plt

# Natural units: GeV
M_P = 1.22e19              # Planck mass (non-reduced), GeV
H0  = 1.5e-42              # H_0 in GeV
rho_obs = M_P**2 * H0**2   # ~ observed Lambda density, GeV^4
rho_naive = M_P**4

# L in GeV^-1
L_horizon = 1.0 / H0
L = np.logspace(np.log10(1e-19), np.log10(1e45), 600)

# CKN bound: rho ≤ M_P^2 / L^2
rho_CKN = M_P**2 / L**2

fig, ax = plt.subplots()

# Naive estimate
ax.axhline(rho_naive, color="#7a3b1f", lw=1.6, ls="--",
           label=r"naïve QFT: $\rho \sim M_P^4$")
ax.text(1e-17, rho_naive*3, r"$\rho_\mathrm{naive} \sim M_P^4$",
        color="#7a3b1f", fontsize=10, fontstyle="italic")

# CKN bound
ax.loglog(L, rho_CKN, color="#a0522d", lw=2.2,
          label=r"CKN bound: $\rho \leq M_P^2 / L^2$")

# Observed
ax.axhline(rho_obs, color="#4f5e48", lw=1.6, ls="-.")
ax.text(1e-17, rho_obs*3, r"$\rho_\Lambda^\mathrm{obs}$",
        color="#4f5e48", fontsize=10, fontstyle="italic")

# Mark Hubble scale
ax.axvline(L_horizon, color="#7a8b6f", lw=0.9, ls=":", alpha=0.85)
ax.text(L_horizon*0.6, 1e-30, r"$L = H_0^{-1}$",
        color="#4f5e48", fontsize=9, fontstyle="italic", rotation=90)

# Mark intersection of CKN with observed
L_intersect = M_P / np.sqrt(rho_obs)
ax.plot(L_intersect, rho_obs, "o", color="#a0522d", markersize=8, zorder=5)
ax.annotate("intersection at $L \\approx H_0^{-1}$",
            xy=(L_intersect, rho_obs),
            xytext=(L_intersect*1e-12, rho_obs*1e8),
            fontsize=9.5, color="#a0522d", fontstyle="italic",
            arrowprops=dict(arrowstyle="->", color="#a0522d", lw=0.8))

ax.set_xlabel(r"region size $L$  (GeV$^{-1}$)")
ax.set_ylabel(r"vacuum energy density  $\rho$  (GeV$^4$)")
ax.set_xlim(1e-19, 1e45)
ax.set_ylim(1e-50, 1e80)
ax.legend(loc="upper right", framealpha=0.9, fontsize=9)
ax.grid(alpha=0.2, which="both")
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

# Print the intersection
print(f"Hubble length L = 1/H_0      ≈ {L_horizon:.2e} GeV^-1")
print(f"CKN-observed crossing scale  ≈ {L_intersect:.2e} GeV^-1")
print(f"ratio                         = {L_intersect/L_horizon:.3f}")

Hubble length L = 1/H_0      ≈ 6.67e+41 GeV^-1
CKN-observed crossing scale  ≈ 6.67e+41 GeV^-1
ratio                         = 1.000

The crossing of the diagonal with the observed value occurs at the scale of the visible universe to within a factor of order unity. This is, on first encounter, a remarkable coincidence. It is the content of the CKN argument.

§3 — What the bound is, and is not

Let us be careful about what Equation 2 establishes and what it does not.

It establishes that the vacuum energy density of an effective field theory consistently defined in a region of size is at most of order . It does not establish that the vacuum energy density is of this order. The bound is an upper bound; it becomes saturated only in regimes where the field theory is, in some sense, maximally populated. There is no general reason, in the present formulation, why a typical patch of vacuum should saturate the bound, and the literature has not — to my knowledge — produced a derivation of saturation from first principles.

What the argument does establish, and what is, I think, its real content, is the following: the discrepancy of between the naïve estimate and the observed value is, in part, an artifact of applying the naïve estimate in a regime where it is not valid. If the IR cutoff is the cosmological horizon, then the UV cutoff of the gravitating vacuum modes is bounded above by approximately , not by the Planck mass. The vacuum modes between and are not absent from the field theory; they are, however, modes whose contribution to gravitating vacuum energy is, by the consistency of the theory, bounded by Equation 1.

The argument therefore does not solve the cosmological constant problem. It rephrases it. The new question — and it is, I believe, a sharper question than the old one — is why the IR cutoff should be the cosmological horizon, and why the bound Equation 3 should be saturated rather than merely respected.

To these questions I have no answer, and the literature has none either; the proposals of Banks and Fischler (2004) and Hsu (2004) are intriguing but not, I think, conclusive. I do not undertake to settle the matter in the present note.

§4 — On the sign

The CKN bound is, as stated in Equation 3, a bound on . It does not, on its face, fix the sign of the result. The observed cosmological constant is positive. Why positive, rather than negative — or zero?

I wish, here, to make a remark which is not, strictly speaking, a derivation, but which seems to me suggestive.

The Casimir effect is the physically measurable consequence of mode redistribution under boundary conditions. For two parallel plates the Casimir energy per unit area is (Casimir 1948), and the resulting force is attractive. For a thin perfectly conducting spherical shell, by contrast, the self-energy is positive (Boyer 1968), by an amount approximately . The two cases are derived from the same formalism. The sign difference arises entirely from the geometry of the boundary.

The cosmological constant problem, on the CKN reading, is a problem about the vacuum energy of a finite region with an IR cutoff. The IR cutoff is, in the cosmological context, a horizon; and a horizon, viewed instantaneously, has the topology of a sphere. The analogy with Boyer’s case is not perfect — the cosmological horizon is not a perfect conducting shell, the modes of the gravitating vacuum are not the modes of the photon field, and the regularization scheme is not the same — but the structural feature responsible for the sign in the Casimir case is the same structural feature one would expect to be operative here: namely, that the vacuum energy of a region bounded by a closed, positively-curved surface is, on geometric grounds, a different quantity, and possibly of a different sign, than the vacuum energy of a region bounded by parallel planes.

If this analogy is to be trusted — and I am not at all certain that it should be trusted to within a factor of order unity, much less to better — then the positivity of the observed cosmological constant may be, in part, a topological consequence of the de Sitter horizon’s being a sphere. The cosmological constant is positive because the universe, in its de Sitter phase, is bounded, in the relevant sense, by a sphere; if it were bounded by parallel planes, it would not be.

I do not advance this as a proof. I advance it as an observation: that the same mathematics which predicts that two plates attract and a sphere repels would, applied with the appropriate care and with attention to the differences between the cases, predict the sign of the cosmological constant. The sign of a result, in Casimir physics, is rarely accidental. Indeed, the geometry will not permit otherwise.

§5 — What survives

I have argued, in outline, the following. The Cohen–Kaplan–Nelson bound Equation 2 is a real and surprising relation between the size of a region and the gravitating vacuum energy it can contain. It does not solve the cosmological constant problem; it sharpens the question to: why is the IR cutoff the cosmological horizon, and why is the bound saturated? The geometric structure of the bound is, I think, consonant with a Casimir-style reading, and the sign of the cosmological constant — its positivity — may be a consequence of the topological character of the de Sitter horizon, in approximately the same sense that the sign of Boyer’s result is a consequence of the spherical character of the conducting shell.

What survives, then, is not a solution but a reorientation. The quantity to be explained is no longer , a number that has no internal structure and resists any natural account. It is, rather, a number that has the form , in which is the cosmological horizon — a quantity that does, on the geometric reading, have an internal structure, and one whose account is now reduced to two more tractable subordinate questions. Whether those questions have natural answers is for those better qualified than I to pursue. The matter is not settled; the matter, on this reading, is at least posed.

References

§2 — On the Several Senses of branching

Before we can profitably engage the question, however, it will be necessary to distinguish, with some care, between several distinct senses in which the term branching is used in the relevant literature, of which I shall identify three, although I am inclined to think — pace the unhelpfully reductive treatment of Stenner (1992) — that a more refined taxonomy might in due course distinguish as many as seven.

The first sense of branching, which I shall call branching-as-Fission, is the sense in which a single antecedent state gives rise, by the unitary evolution of the joint system-and-apparatus, to two posterior states, each of which is equally a continuation of the antecedent state. This sense is, broadly speaking, the sense employed by Lewis (1976) in the analogous case of personal fission, and it has been imported into the quantum case by, inter alia, Pemberton-Hughes (1987) — though, it must be said, with insufficient attention to the differences between the modal structure of the macroscopic and the quantum cases, a point to which we shall return in §4.

The second sense, which I shall call branching-as-Multiplication, treats the resulting branches as numerically distinct individuals, each of which has come into being at the moment of branching, and neither of which is to be identified with the antecedent. This is, I take it, the sense most natural to the de dicto reading of identity-claims; whether it is the right reading is a matter to which we shall return.

The third sense — and the most subtle — is what I shall call branching-as-Selection: on this view, only one of the resulting branches is, strictly speaking, the continuation of the antecedent, the others being, in some sense to be made out, modal residues of the original individual. This is the sense whose articulation is the principal task of the present note, and to which, accordingly, we now turn.

The careful reader will observe that the three senses are not, on their face, mutually consistent; this is not, however, a defect of the taxonomy but a feature of the underlying problem, which has been obscured — I am sorry to say — by the loose usage of the term “branching” in the recent literature.

§3 — The De Re Reading of Branched-Identity-qua-Identity

Let us suppose, for the sake of definiteness, that an individual A, prior to a branching event, possesses the de re property F. The question we wish to ask is: of the two posterior individuals, A_1 and A_2, which (if either, or neither, or both) possesses the property F in the same sense in which A possessed it?

It is, I think, instructive to consider three possible answers.

The first answer — which I shall call the Quasi-Lewisian answer, after the analogous treatment in Lewis (1976) — is that both A_1 and A_2 possess F, and that each does so in exactly the same sense in which A possessed it; on this view, the property F has, as it were, propagated forward into both branches, salva veritate. This answer has the merit of simplicity. It has, however, in my view, the disadvantage that it leaves wholly unaddressed the modal aspect of F, which is to say: the question of whether the F possessed by A_1 is the same F possessed by A, or merely a qualitative duplicate of it — a distinction that, I take it, no careful philosopher of language would wish to elide.

The second answer is that neither A_1 nor A_2 possesses F in the sense in which A possessed it, since A has, by the branching event, ceased to exist, and the property F, having lost its bearer, has accordingly become — in a phrase I shall borrow, with thanks, from Vaihinger (1911) — a fictive predicate. This answer, while not without its merits, requires us to accept a metaphysical conclusion of considerable strength, namely the non-identity of the post-branching individuals with the antecedent, and I confess I am unable to do so without further argument.

The third answer — which I shall defend — is that the property F, as possessed by A, is neither possessed by A_1 in the same sense, nor possessed by A_2 in the same sense, nor possessed by both in some compound sense, nor lost altogether: it is, rather, modally distributed across the two branches in a manner that the existing literature has not adequately characterized. The unaddressed remainder, when the standard taxonomy is applied to the case, is what I shall call the modal residue.

§4 — The Modal-Residue Problem

The modal residue may be characterized, with some attempt at precision, as follows. Let A_1 and A_2 be the two posterior individuals, and let f_1 and f_2 denote the post-branching counterparts of the property F — that is, the properties which stand to F as A_1 and A_2 stand, respectively, to A. Then the modal residue R is defined by

— here I lapse, with apology to the more mathematically inclined among my readers, into a notation borrowed from Pemberton-Hughes (1987), adapted (with some technical modifications) for the present case —

where ⊕ denotes the modal sum of the post-branching properties, in the sense due to Stenner (1992). The quantity R, which is in general non-zero, represents the portion of the antecedent property F that has not been recovered by either of the post-branching counterparts, and which therefore, if the analysis is to be coherent, must be assigned a place somewhere in the modal landscape. The question — and it is a question, I think, of the first importance — is where.

It will be objected, no doubt, that I have not yet specified what the modal sum ⊕ is supposed to be, nor in what space the quantity R of Equation 1 is to be regarded as living. I shall, in §5, argue that these subsidiary questions have been adequately addressed in the literature — though, again, only by careful philosophers — and that the principal remaining task is the application of the resulting machinery to the case at hand. For the moment, however, I wish only to insist on the following: that the modal residue is a real quantity, that it is in general non-zero, and that any account of the Everett interpretation which does not provide for its placement leaves the metaphysics of the situation in a condition that no careful reader can find satisfactory.

The matter is, I think it must be said, somewhat surprising. One would have expected the philosophical community, in the four decades since Everett (1957), to have arrived at a settled view on the placement of R; and that this has not happened — that, on the contrary, the question has not, so far as I can determine, been posed in the form I have given it — is itself a fact about the recent literature that deserves comment, though I shall not here undertake to make that comment, salva veritate, in the form it deserves.

§5 — The Inadequacy of the Existing Treatments

The recent literature on the Everett interpretation — which is to say, broadly, the literature that has accumulated since DeWitt (1970) and the early commentaries — has not, on the whole, attended to the modal-residue problem in the form in which I have set it out, and has accordingly, I think it is fair to say, fallen short of providing a complete account of the metaphysics of branching. Several treatments deserve mention.

Stenner (1992), in a much-overlooked monograph, set out a notion of modal aggregation that, while not framed in the terms I have used here, is structurally close to what I have called modal sum; it would, I believe, repay further attention from those willing to follow the argument through the seventh chapter, where the notation, on first encounter, is admittedly somewhat forbidding. Pemberton-Hughes (1987), who introduced the notation I have above adopted, applied it — owing, I think, to insufficient familiarity with the de re / de dicto distinction — to a question rather different from the one I have addressed, with the result that his conclusions, while suggestive, are prima facie unsuited to the present case.

The recent decision-theoretic literature on the so-called “probability problem” of the Everett interpretation, of which I am aware in outline, has not, to my knowledge, addressed the modal-residue question at all; and while I have no objection in principle to decision-theoretic methods — they have, in their proper sphere, an unimpeachable distinguished history — I am inclined to think that, in the present context, they have been deployed in pursuit of the wrong question. The probability problem, on the analysis I have offered, is a consequence of the modal-residue problem, rather than an independent question, and its treatment in advance of the prior question is therefore, in a precise sense, putting the cart before the horse. The proper order of investigation is: first the residue, then the probability; the literature has, regrettably, proceeded in the reverse order.

A treatment that does not address the modal residue will, a fortiori, be unable to distinguish — for reasons I shall not here elaborate, though I have done so at greater length in correspondence — between the Quasi-Lewisian, the fictive, and the modally distributed readings, and will accordingly produce conclusions about the metaphysics of branching that are not, strictly speaking, conclusions at all, but rather prima facie claims awaiting the further analysis I have undertaken to provide.

§6 — The Pemberton-Hughes Move: A Reply

It will be objected by some — and the objection has been made to me, in correspondence, with characteristic vigor — that the modal-residue problem has, in fact, been addressed, by Pemberton-Hughes (1987) and the small subsequent tradition, in the form of what is now sometimes called the Pemberton-Hughes Move: namely, the proposal that the modal residue is, in all cases of physical interest, identically zero, on the grounds that the properties f_1 and f_2 together exhaust the modal content of the antecedent property F, leaving, ex hypothesi, no remainder.

I am not persuaded.

The Pemberton-Hughes Move depends, in essence, on the assumption that the modal sum ⊕ is, in the relevant cases, complete — an assumption that, on close inspection, is equivalent to the very claim that there is no modal residue, and which is therefore in danger of begging the question it purports to settle. I have argued elsewhere — though without, I should concede, the technical apparatus that the present note provides — that the assumption of completeness is, in general, unwarranted, and that the residue is, in cases of philosophical interest, non-trivially non-zero. The Pemberton-Hughes Move thus collapses, as a defense of completeness, into a restatement of the position it was meant to defend; the collapse is, I think, complete in a sense more interesting than the one Pemberton-Hughes intended.

A further reply has been suggested to me — again in correspondence — to the effect that one might dispense with the modal sum altogether, and treat the post-branching individuals as fully constituted in their own right, with no residue to be placed. This reply, while ingenious, runs aground on the de re aspect of the property F in the manner I have already indicated in §3, and I shall not here belabor the point.

§7 — Resolution; or, Where the Matter Stands

I have argued, in outline, the following: that the question of identity in the Everett interpretation has been treated, in the existing literature, with insufficient attention to the de re / de dicto distinction; that the de re reading, properly developed, gives rise to what I have called the modal-residue problem, of which the principal mathematical content is captured by Equation 1; and that no extant treatment of the Everett interpretation provides an adequate placement of the modal residue, with the consequence that the standard accounts are, in a precise sense, incomplete.

What, then, is the upshot? I am inclined, on balance, and pace the suggestion of Pemberton-Hughes (1987) to the contrary, to think that the modal residue is to be located in what one might call the intermediate modal space between the two post-branching branches — that is, in the region whose ontological status has, in the past, been variously denied, ignored, or fictively assigned, but whose existence is, on the present account, both necessary and metaphysically respectable. The branches are, on this view, not the whole of the post-branching reality; they are, rather, two regions of a larger structure, the remaining portion of which is the modal residue. The Everett interpretation, properly understood, is therefore not a two-branch theory but a branched-with-residue theory — a position whose implications I shall not, in the present note, attempt to develop, but which I commend to the attention of those interested in the foundations of quantum mechanics, and which I shall pursue at greater length in a forthcoming monograph.

The matter is, I take it, on close examination, settled — though not, perhaps, in the sense that has been ordinarily assumed.

I look forward to correspondence with my colleagues at the Institute on these and related matters, and to my next visit, in the autumn.

— L. Bristlecone

Editorial responses

The “modal residue” introduced in §4 has, so far as I can establish, no representation in the Hilbert space on which the formalism of Everett (1957) is defined; the operation ⊕ is similarly unspecified in any sense that an inner product would recognize. The canonical interpretive question — the recovery of the Born rule — is not engaged in the present note.

— M. Yucca, 22 September 2003

L. Bristlecone has, with characteristic care, set out a problem and offered a resolution. I read the note with attention, on the porch, after supper. I find I have one small question, which I shall pose in the simplest terms I can manage.

Suppose, for a moment, that I were to grant the modal-residue problem of branched-identity-qua-identity in the form set out in §4, and to grant further that the Pemberton-Hughes Move (§6) is to be rejected for the reasons given. The matter would then stand, as L. Bristlecone has it, with the residue placed in the intermediate modal space between the branches. Now suppose, alternatively, that I were to grant the Pemberton-Hughes Move and to deny the existence of any non-trivial residue. The matter would then stand otherwise.

My question is this: what experimental fact, or observation, or calculation — even one in principle, even one I cannot now specify but might in some imagined laboratory carry out — would come out differently between the two cases? I ask in good faith. If the answer is that no such fact exists, then we are in a region of philosophy whose problems we at the Institute have, by long experience, found do not yield well to further elaboration. The region contains real problems and unreal ones, and I am not always sure I can tell them apart from the porch.

In any event, a glass has been set aside, against your next visit, for the productive disagreement we shall by then have had. The cottonwoods are coming on; you would, I think, find the property pleasant in October.

— D. R. Caldera, 1 October 2003

References

The mode sum

Between the plates, the electromagnetic field is most cleanly expanded in the modes that satisfy the boundary conditions. For each pair of transverse wavevectors and longitudinal mode index , there are two polarizations (with contributing only one), and the angular frequency of the corresponding mode is

The zero-point energy per unit plate area, formally, is the sum

with the prime indicating that the term is to be counted with weight , on account of its single polarization.

The sum Equation 3 is, of course, divergent. This is where the Casimir literature begins, and where most introductory accounts pause to apologize. I do not think an apology is in order. The divergence is not a defect of the calculation; it is a feature of the formal vacuum energy, which is unphysical in absolute terms and becomes physical only in differences. The physical quantity is

where is the same sum performed with the plates removed — that is, with the discrete index replaced by a continuous integral. The two divergences cancel, and what remains is finite.

To extract the finite part we regularize, as is conventional, by introducing a smooth cutoff function which is unity for and falls off rapidly for . The choice of does not affect the result. One could equally well analytically continue, in the manner of Plunien et al. (1986), using the Riemann zeta function to assign a finite value to the bare sum; the answer is the same. The cancellation is exact, and on first encounter is surprising.

The regularized result

Substituting Equation 2 into Equation 3, performing the angular part of the integral, and writing , the regularized energy per unit area takes the form

The corresponding integral with continuous is the Euler–Maclaurin “smooth” piece, and the difference between the discrete sum and the integral is the finite Casimir energy. The Euler–Maclaurin formula gives

independent of in the limit . The attractive pressure Equation 1 follows as .

Numerical evaluation

import numpy as np
import matplotlib.pyplot as plt

hbar = 1.055e-34
c = 3.0e8
a = np.logspace(-9, -4, 600)  # plate separation in meters
P = -(np.pi**2 / 240) * hbar * c / a**4   # pressure (Pa), negative = attractive

fig, ax = plt.subplots()
ax.loglog(a*1e6, -P, color="#a0522d", lw=2.0,
          label=r"$|P| = \pi^2 \hbar c / 240 a^4$")

# Reference pressures
ax.axhline(101325, color="#7a8b6f", lw=0.8, ls=":", alpha=0.7)
ax.text(1.2e-3, 1.5e5, "1 atm", color="#4f5e48", fontsize=8.5, fontstyle="italic")
ax.axhline(1.0, color="#7a8b6f", lw=0.8, ls=":", alpha=0.7)
ax.text(1.2e-3, 1.5, "1 Pa", color="#4f5e48", fontsize=8.5, fontstyle="italic")

# Lamoreaux 1997 range
ax.axvspan(0.6, 6.0, color="#d9b382", alpha=0.35,
           label="Lamoreaux 1997 measurement")

# Annotations
for a_val, label in [(1e-2, "1 μm"), (1e-1, "100 nm"), (1e-3, "10 nm")]:
    pass

ax.set_xlabel(r"plate separation $a$ ($\mu$m)")
ax.set_ylabel(r"|Casimir pressure| (Pa)")
ax.set_xlim(1e-3, 1e2)
ax.legend(loc="upper right", framealpha=0.9, fontsize=9)
ax.grid(alpha=0.25, which="both")
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

# Specific values
for a_val in [10e-9, 100e-9, 1e-6, 10e-6]:
    P_val = (np.pi**2/240) * hbar * c / a_val**4
    print(f"a = {a_val*1e9:7.1f} nm   |P| = {P_val:.3e} Pa")

a =    10.0 nm   |P| = 1.302e+05 Pa
a =   100.0 nm   |P| = 1.302e+01 Pa
a =  1000.0 nm   |P| = 1.302e-03 Pa
a = 10000.0 nm   |P| = 1.302e-07 Pa

The pressure is small at any separation a careful experimentalist would call macroscopic. At , — about ten thousand times smaller than a millibar of atmospheric pressure. The first credible measurement of Equation 1 in this regime is due to Lamoreaux (1997), who used a torsion pendulum with a spherical lens replacing one of the plates (the sphere-plate geometry being mechanically more tractable, and the geometry having since been treated more carefully than I shall do here, in the proximity-force approximation and beyond). The agreement with theory was within , which is small for this kind of experiment.

At the pressure is approximately , comparable to atmospheric. This is the regime in which the Casimir force becomes structurally relevant in the design of microelectromechanical devices (Chan et al. 2001); it is, in a quite practical sense, what a small enough machine has to push against when it tries to move two of its parts together.

What the geometry has done

I should like to close on the geometry, where I began. The result Equation 1 depends on , , and the separation — and on nothing else. It does not depend on the material of the plates, provided the plates are good enough conductors at the relevant frequencies. It does not depend on the area , except trivially through the conversion from energy to pressure. It does not even depend on the choice of regularization. What it depends on is the shape of the boundary: two parallel planes, an infinite gap in transverse coordinates, a finite gap in the third.

A different shape would give a different sign, in some cases. The case of a thin perfectly conducting sphere, treated by Boyer (1968), yields a repulsive self-energy, of magnitude smaller by some four orders of magnitude than the parallel-plate value at the corresponding scale. The cancellations there are more delicate, and the result was, on first encounter in 1968, badly received. It is now, I believe, accepted. Indeed, the geometry will not permit otherwise.

References

Boyer, Timothy H. 1968. “Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle.” Phys. Rev. 174: 1764–76.

Casimir, H. B. G. 1948. “On the Attraction Between Two Perfectly Conducting Plates.” Proc. K. Ned. Akad. Wet. 51: 793–95.

Chan, H. B., V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso. 2001. “Quantum Mechanical Actuation of Microelectromechanical Systems by the Casimir Force.” Science 291: 1941–44.

Lamoreaux, S. K. 1997. “Demonstration of the Casimir Force in the 0.6 to 6 m Range.” Phys. Rev. Lett. 78: 5–8.

Plunien, G., B. Müller, and W. Greiner. 1986. “The Casimir Effect.” Phys. Rep. 134: 87–193.

Part one: Pauli’s theorem

The Page–Wootters construction requires a clock subsystem whose Hamiltonian admits a self-adjoint canonical conjugate operator with continuous spectrum running over all of . The conditional state is then defined, as in equation (3) of Caldera (1986), as the projection of the joint state onto an eigenstate of .

Pauli (1933) showed, in a footnote that has been more cited than read, that no such operator can exist if is bounded below. The argument is short. Suppose . Then for any energy eigenstate and any real ,

so the spectrum of contains for every real , contradicting any lower bound. A self-adjoint therefore requires a Hamiltonian unbounded below; no physical clock has this.

The remedies are known. One may replace the projection-valued measure of with a positive operator-valued measure (POVM); one may extend the Hilbert space to admit “covariant” time observables; one may interpret the construction in the sense of Rovelli (1991) as a relational specification of partial observables. Each of these remedies is technical and each modifies the conditional probability in ways that the Director’s exposition would not have admitted without footnoting. The mechanism, in its naive formulation, is therefore an idealization.

Part two: The Salecker–Wigner bound

Setting Pauli aside, suppose one grants the existence of a workable clock observable. There remains a quantitative limit on the precision with which any physically realizable clock — meaning: any system of finite mass and finite energy — can resolve a time interval. The limit was derived by Salecker and Wigner (1958) and refined by Wigner (1957). It states that a clock of total mass which is to operate, as a clock, for a total elapsed time , has a minimum resolvable time interval

The result follows from elementary uncertainty arguments: a clock of energy uncertainty has a time resolution ; over a total operating time the wavefunction of a clock of mass spreads spatially by at least ; combining these, with , yields Equation 2.

The bound is not a function of the clock’s design. It is a consequence of quantum mechanics together with the finiteness of the clock’s mass.

Numerical evaluation

Let us substitute the numbers, as is the habit of the house.

import numpy as np
import matplotlib.pyplot as plt

hbar = 1.055e-34
c = 3.0e8
T = 365.25 * 24 * 3600   # 1 year, in seconds

M = np.logspace(-30, 5, 600)   # mass in kg, from atomic to a small house
dt_min = np.sqrt(hbar * T / (M * c**2))

fig, ax = plt.subplots()
ax.loglog(M, dt_min, color="#2b2118", lw=2.0)

events = [
    ("electron",                    9.11e-31, "#7a8b6f"),
    ("Cs-133 atom",                 2.21e-25, "#7a8b6f"),
    ("a brass railroad clock",      2.0,      "#a0522d"),
    ("the Director's pickup truck", 2.0e3,    "#7a3b1f"),
    ("Planck mass",                 2.18e-8,  "#b8794f"),
]

for label, M_val, color in events:
    dt_val = np.sqrt(hbar * T / (M_val * c**2))
    ax.plot(M_val, dt_val, "o", color=color, markersize=6, zorder=5)
    ax.annotate(label, xy=(M_val, dt_val),
                xytext=(M_val*1.7, dt_val*0.25),
                fontsize=8.5, color=color, fontstyle="italic")

# Reference: Cs-133 atomic clock published precision ~1e-15 s
ax.axhline(1e-15, color="#4f5e48", lw=0.8, ls=":", alpha=0.7)
ax.text(1e-30, 3e-15, "Cs-133 clock published $\\Delta t$",
        fontsize=8.5, color="#4f5e48", fontstyle="italic")

ax.set_xlabel(r"clock mass $M$ (kg)")
ax.set_ylabel(r"minimum $\Delta t$ over $T = 1$ yr  (s)")
ax.grid(alpha=0.25, which="both")
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

For the brass railroad clock the bound is , which is far below the resolution of the hands and below any timescale of seminar interest. For an atomic clock the bound is approximately , which is within an order of magnitude of the published precision of caesium references and is the source of the long-running interest in heavier-atom standards.

The bound is not in conflict with any practical use of clocks. It is, however, in conflict with the idealization that the clock in a Page–Wootters universe can be made arbitrarily precise without also being made arbitrarily massive. Indeed, by the time the clock has acquired enough mass to track time at any chosen precision over cosmological intervals, it has acquired enough mass to disturb, gravitationally, the system which it is supposed to be correlated with but not to perturb (Hartle 1988).

A short conclusion

The Page–Wootters mechanism is a useful and probably correct picture of how time emerges in a closed quantum universe. It is not a mechanism that operates without cost. The cost is paid in the physical clock — in its mass, its precision, and its gravitational coupling — and the cost has been carefully bounded by classical results (Pauli (1933), Salecker and Wigner (1958), Wigner (1957)) which the proposal does not modify.

I make this remark not to disagree with the Director, with whom I agree on the essentials, but because at the Institute we have made something of a habit, when proposing that an apparent difficulty has been resolved, of asking what the resolution costs. The answer here is small and bounded, but it is not zero.

References

Caldera, D. R. 1986. “The Wheeler–DeWitt Equation and the Disappearance of .” Notes & Preprints, Chuck Walla Institute.

Hartle, James B. 1988. “Quantum Mechanics of Cosmological Observations.” Phys. Rev. D 37: 2818–32.

Page, Don N., and William K. Wootters. 1983. “Evolution Without Evolution: Dynamics Described by Stationary Observables.” Phys. Rev. D 27: 2885–92.

Pauli, Wolfgang. 1933. “Die allgemeinen Prinzipien der Wellenmechanik.” Handbuch Der Physik 24: 83–272.

Rovelli, Carlo. 1991. “Time in Quantum Gravity: An Hypothesis.” Phys. Rev. D 43: 442–56.

Salecker, H., and E. P. Wigner. 1958. “Quantum Limitations of the Measurement of Space-Time Distances.” Phys. Rev. 109: 571–77.

Wigner, E. P. 1957. “Relativistic Invariance and Quantum Phenomena.” Rev. Mod. Phys. 29: 255–68.

What the eye is doing

The eye, at this hour, is performing a long and difficult act of integration. It is collecting photons from the sun — most of them from a region between perhaps and astronomical units — that have been redirected by single scattering off the surfaces of dust grains and that arrive at the dark-adapted observer at intensities between perhaps and units, where one is the surface brightness of one tenth-magnitude star per square degree. The total integrated brightness of the zodiacal light, summed over the sky, is comparable to that of all the unresolved stars in the Milky Way put together (Reach 1996).

The intensity, as a function of ecliptic latitude and elongation from the sun, is well-fit at small ecliptic latitudes by an empirical relation of the form

with , applicable in the range where the inner-zodiacal physics is not the dominant contribution (Leinert et al. 1998). The form Equation 1 encodes two facts: that the dust is concentrated near the ecliptic plane, and that the column density falls off with distance from the sun roughly as .

import numpy as np
import matplotlib.pyplot as plt

beta = np.linspace(-90, 90, 600)
beta0 = 30.0
I = (1 + (beta/beta0)**2)**(-1.5)
I_full = 220 * I  # in S_10 units, schematic normalization at ecliptic

fig, ax = plt.subplots()
ax.plot(beta, I_full, color="#a0522d", lw=2.0)
ax.fill_between(beta, 0, I_full, color="#d9b382", alpha=0.35)

# Ecliptic and pole markers
ax.axvline(0, color="#7a3b1f", lw=0.8, ls=":", alpha=0.7)
ax.text(2, 230, "ecliptic", color="#7a3b1f", fontsize=9, fontstyle="italic")
ax.axvline(90, color="#7a8b6f", lw=0.8, ls=":", alpha=0.6)
ax.axvline(-90, color="#7a8b6f", lw=0.8, ls=":", alpha=0.6)
ax.text(78, 60, "ecliptic\npole", color="#4f5e48", fontsize=8.5, fontstyle="italic")

ax.set_xlabel(r"ecliptic latitude  $\beta$  (degrees)")
ax.set_ylabel(r"surface brightness  ($S_{10}$ units)")
ax.set_xlim(-90, 90)
ax.set_ylim(0, 280)
ax.grid(alpha=0.2)
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

The slow falloff with is what gives the cone its considerable breadth at the horizon. Were the dust confined to a thin disk, the zodiacal light would be a narrow ribbon; instead, it is a pyramid.

The morning in question

The morning these notes record was the eighth of March, 1992. The moon had set at half past one. The sky was clear in the way that desert skies are clear after a windy day, with the dust settled and the air dry. I drove south on Last Chance Road, parked where the washboard becomes uncomfortable, and waited on the hood with a thermos and an old Brunton compass.

The cone came up at about a quarter past five. It was visible to the naked eye for perhaps forty minutes before astronomical twilight began to wash it out. Below the cone, just at the horizon, I caught once or twice the brighter inner gegenschein — the diffuse backscattering from dust at the antisolar point, an entirely separate phenomenon and the harder one to see — though the geometry was wrong for a clean detection.

A pair of bats, as is usual at that hour, came past low and turned inland.

What is being measured

The zodiacal light is not, in cosmological terms, important. The extragalactic background, which I have written about elsewhere (Mesquite 1988), is fainter by orders of magnitude. The scientific interest of the zodiacal light is principally that it is a foreground: to measure the diffuse extragalactic background light from any satellite mission whose orbit crosses the ecliptic plane — which is most of them — one must subtract the zodiacal contribution to better than a percent. The DIRBE instrument aboard COBE did this work patiently between 1989 and 1990 (Hauser et al. 1998), and the careful zodiacal model (Kelsall et al. 1998) that resulted is, in some quiet sense, the price one pays for having seen anything at all of the sky beyond.

But the cone, observed from the desert, is not a foreground to anyone in particular. It is what a few billion tonnes of cometary debris look like in the right light. The grains, individually, are smaller than one can see; the cloud, integrated, is the second-brightest diffuse object in the inner solar system. We who live where the air is dark are the people who get to see it.

A small closing note

A naked eye can register the zodiacal light from perhaps three hundred sites in the lower forty-eight states. Last Chance Road is one of them. I would urge the visitor — particularly the visitor arriving in autumn, when the morning ecliptic stands steepest — to rise an hour earlier than is comfortable, walk south of the Institute, and look east. The pyramid is patient. It will wait for the eye to adapt.

References

The static-universe argument

Consider a universe of infinite extent, in which stars of mean luminosity are distributed with uniform number density , and in which the geometry is Euclidean and static. Consider a thin spherical shell of radius and thickness centered on the observer. The shell contains stars, each delivering a flux at the observer of . The total flux from the shell is therefore

The two factors of cancel: every shell contributes equally, regardless of its distance. Integrating from the observer outward,

The night sky should be infinitely bright. Even granting that nearer stars eventually occlude more distant ones, a thermodynamic re-derivation due to Harrison (1987) shows that in steady state the sky must reach the surface temperature of the average star — some thousands of kelvin. This is unambiguously not what one observes from Last Chance Road.

The resolutions, in order of importance

The correct response is twofold and was, by the 1960s, broadly settled (Harrison 1987; Wesson 1991):

1. The universe has a finite age. Only stars within a light-travel distance of approximately light-years can have contributed photons to tonight’s sky. The integral Equation 2 is therefore cut off at finite .
2. The universe is expanding. Photons from distant stars arrive redshifted, with their energies reduced by a factor and their arrival rate reduced by a further . The intensity from a shell at redshift is therefore suppressed by .

The first effect is the dominant one. The second is, on its own, insufficient to resolve the paradox; Harrison (1987) makes this point emphatically. But together they reduce the integrated extragalactic background light to a value many orders of magnitude below the surface brightness of the Milky Way’s foreground stars, which is what the eye actually sees as it adapts in the dark.

Numerics

We compute the cumulative surface brightness of the night sky as a function of the cutoff distance, comparing the static-Euclidean case Equation 2 to a finite-age, expanding model.

import numpy as np
import matplotlib.pyplot as plt

# Comoving distances in Gly
r = np.linspace(0.001, 80, 4000)

# Static infinite: F(r) ~ r (linear, diverges)
F_static = r.copy()

# Toy expanding-universe: integrand = exp(-r/14) * (1+z)^{-4}
# with redshift approximated by Hubble: z ≈ H0 r / c, so for r in Gly with t_0=14 Gly,
# z ≈ r / (c/H0) ~ r / 14 Gly.
t0 = 14.0  # Hubble distance in Gly (rough)
z_of_r = r / t0
suppression = (1 + z_of_r)**(-4) * np.exp(-r/t0)  # crude horizon cutoff
F_expand = np.cumsum(suppression) * (r[1]-r[0])

fig, ax = plt.subplots()
ax.plot(r, F_static, color="#2b2118", lw=1.8, label="static, infinite (Olbers)")
ax.plot(r, F_expand, color="#a0522d", lw=2.2, label=r"expanding, finite-age ($\Lambda$CDM, schematic)")
ax.axvline(t0, color="#7a8b6f", ls=":", lw=1.2, alpha=0.8)
ax.text(t0+1, F_expand[-1]*0.45, "Hubble distance",
        color="#4f5e48", fontsize=9, fontstyle="italic")
ax.set_xlabel("cutoff distance $r$ (Gly)")
ax.set_ylabel("integrated flux  (arb. units)")
ax.set_xlim(0, 80)
ax.legend(loc="upper left", framealpha=0.9)
ax.grid(alpha=0.2)
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

The plateau in the rust-colored curve is the resolution of the paradox. A finite-age, expanding universe deposits a finite photon density on the night sky.

What one actually sees from Last Chance Road

The integrated extragalactic background, once we account for the factors above, is approximately — a number many orders of magnitude below the threshold of dark-adapted human vision. What the eye sees in a truly dark sky (the Bortle 1 of amateur astronomers, attainable with some effort in the desert Southwest) is, in order of decreasing brightness:

1. Airglow — recombination in the upper atmosphere — at perhaps .
2. The integrated light of unresolved Milky Way stars.
3. Zodiacal light from interplanetary dust.
4. Faintly, on the very darkest nights, the diffuse light from unresolved galaxies — an experimentally measurable quantity, as verified now by COBE’s DIRBE (Hauser et al. 1998).
5. Below all of these, the cosmic microwave background — invisible to the eye, but present everywhere on the sky at .

The CMB is, in a precise sense, the answer to Olbers: it is the hot wall predicted by the static-universe argument, redshifted by the expansion from a few thousand kelvin down to its present three. The night sky is dark because the universe expanded.

A closing observation

The Institute’s outdoor blackboard, on the evening in question, remained legible by starlight alone for some forty minutes after the last lantern was extinguished. The legibility is, when one thinks about it, a quietly extraordinary fact: it depends on the Hubble parameter, the matter content of the universe, and the surface temperature of G-type stars three to four kiloparsecs distant. We mention this not as a discovery, but to suggest that some of the Institute’s better seminars take place out of doors.

References

What is missing, and why

In ordinary quantum mechanics the Schrödinger equation has the form

and the parameter is, however unscrutinized, there. It is the parameter against which the state evolves. The wavefunction at is something; the wavefunction at is something related; the relation is provided by Equation 2, which presupposes that there is a to differentiate with respect to.

In Equation 1 there is no such parameter. This is not a notational omission. The reasoning that produces Equation 1 — diffeomorphism invariance of the underlying gravitational theory — requires that no preferred time function appear. The universe, at its most fundamental level, does not evolve. It simply is, in some configuration that satisfies the constraint.

This is the problem of time, as canvassed at length by Kuchař (1991) and Isham (1992). The literature is large and the resolutions are many; none has won general assent.

The Page–Wootters proposal

The proposal due to Page and Wootters (1983), called by them evolution without evolution, is the one that has lodged most firmly in my own thinking. Their argument may be put as follows.

Consider a closed quantum system — a universe — composed of two subsystems, which we shall call and . Suppose the joint state satisfies the timeless constraint

There is, by hypothesis, no external time. Now suppose that the state is correlated in the following sense: the subsystem is what we shall call a clock, meaning that its Hamiltonian has a self-adjoint canonical conjugate — a position-like operator with a continuous spectrum. Then the conditional state of , given that the clock has been observed to read the value , is

A short calculation, using Equation 3, shows that this conditional state satisfies

That is: the timeless state, conditioned on the clock’s reading, looks to subsystem exactly like the standard time-evolved quantum state of Equation 2. The parameter which appeared to be missing has reappeared, not as a fundamental variable, but as the value of one subsystem’s reading correlated with the state of another.

A picture of two clocks

import numpy as np
import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 3, figsize=(7.6, 2.8))

# A series of three "snapshots" — each conditioned on a different reading of clock B
ts = [0.0, 1.2, 2.4]
for ax, t in zip(axes, ts):
    # Clock A: hand at angle proportional to t
    theta_A = -np.pi/2 + 1.3 * t
    ax.add_patch(plt.Circle((0, 0), 1.0, fill=False, color="#7a3b1f", lw=1.6))
    ax.plot([0, 0.7*np.cos(theta_A)], [0, 0.7*np.sin(theta_A)],
            color="#a0522d", lw=2.2)
    # Clock B's reading shown below as a dial reading
    ax.text(0, -1.6, f"clock $B$ reads $t = {t:.1f}$",
            ha="center", fontsize=10, color="#2b2118", fontstyle="italic")
    ax.text(0, 1.25, "clock $A$", ha="center", fontsize=10,
            color="#7a3b1f", fontstyle="italic")
    # Hour ticks
    for k in range(12):
        a = 2*np.pi * k / 12
        ax.plot([0.92*np.cos(a), 1.0*np.cos(a)],
                [0.92*np.sin(a), 1.0*np.sin(a)],
                color="#7a3b1f", lw=0.9)
    ax.set_xlim(-1.4, 1.4)
    ax.set_ylim(-2.0, 1.5)
    ax.set_aspect("equal")
    ax.axis("off")
    ax.set_facecolor("#faf3e3")

fig.patch.set_facecolor("#faf3e3")
fig.suptitle(r"$|\psi_A(t)\rangle \ = \ \langle t|_B\,|\Psi\rangle_{AB}$",
             fontsize=11, color="#2b2118", y=0.04)
plt.tight_layout()
plt.show()

The figure shows what the proposal asks one to imagine. There is no animation in the underlying state — is, by Equation 3, an eigenstate of zero energy and does not change. The appearance of a moving hand on clock is recovered when we look at the conditional state given each successive reading of clock . What looks like time is a property of the correlation between the two subsystems.

It is, I think, a quietly remarkable proposal. It does not solve the problem of time in the sense that any cosmologist would call solved; the construction is technically delicate, the decomposition of the universe into “clock” and “rest” is not unique, and the recovery of the standard semiclassical limit involves choices that the literature has not finished arguing over. But it does suggest that the disappearance of from the fundamental equation is not a defect of the theory. Time is a relation, not a substance; the equation omits it as one might omit, from a description of a marriage, the exterior weather.

What the two clocks at the Institute have been doing

The brass clock in the seminar room and the electric clock in the kitchen do not, of course, satisfy a Wheeler–DeWitt constraint. Their correlations are mediated by a Hamiltonian that contains, among other things, the local power grid and the temperature of the brass case. But the structure of the situation is not different from the cosmological case in any deep way. Each clock measures the other. What we call time, when we glance at one and then at the other and say “the brass one is slow,” is a fact about how their readings are correlated. It is not a fact about a separately existing parameter which both are imperfectly tracking.

I have, since 1981, found this thought consoling. The clocks need not agree to keep good time. They need only keep correlated time. The difference matters principally to the physicist; for the rest of the Institute, breakfast occurs when the kitchen clock says it does, and the seminar begins, more or less, when the brass clock chimes.

A closing observation

The chuckwalla, Sauromalus ater, does not keep time. He notices the sun, and the temperature of the rocks, and the relative positions of the morning shadows; from these he assembles a record adequate to his needs. We are, in the relevant sense, doing the same. We have more clocks, and the corrections are more careful, and we sometimes write papers about the corrections; but the underlying activity is the old one, of inferring the state of one part of the world from the reading of another.

The brass clock has just struck, and although it is slow by perhaps a minute, the kitchen agrees that it is time to put the kettle on.

References

The relativity of simultaneity, briefly

Consider two observers, and , in inertial motion relative to one another with velocity along the -axis. The Lorentz transformation relates their coordinates by

Two events at the same time in ’s frame, separated by spatial distance , are not simultaneous in ’s frame; rather, they are offset by

The relation Equation 2 is the entire content of what we shall need.

A numerical illustration, from the porch

Let be an observer seated, motionless, on the porch of the Institute at Six Mile Spring. Let be the driver of an eastbound pickup truck on Last Chance Road, who has just turned onto Nevada State Route 160 and is headed toward Las Vegas at the legal limit of fifty-five miles per hour, or . The distance from the Spring to the Las Vegas Strip, by way of Pahrump and the Spring Mountains, is approximately .

The simultaneity offset Equation 2 evaluates, with the relevant numbers, to

or thirty picoseconds. The driver’s now, as it slices through the Strip, is offset by thirty picoseconds from the now of the porch. This is, by any reasonable standard, an unimpressive quantity. The point is not the magnitude.

The point is that the offset is non-zero, and that it depends on the driver’s velocity and on the direction in which she happens to be looking. Were she to make a U-turn, her plane of simultaneity would tilt the other way; were she to accelerate to a higher speed, it would tilt further. There is, in the geometry, no preferred slicing of spacetime into simultaneous events; there are only the slicings that particular observers happen to adopt.

The Andromeda generalization

The argument is sometimes felt to acquire its force only when extended to larger distances. Following the original presentation: replace Las Vegas with the Andromeda galaxy, at a proper distance of light-years, and replace the pickup truck with a leisurely walk in the direction of Andromeda at . Then

which is three days. If one elects to walk toward Andromeda, the events on Andromeda which one regards as “happening now” are offset by three days relative to the events one regarded as “happening now” while standing still. A decision to take a step has, on this reading, an immediate consequence for what is real in a galaxy two and a half million light-years away. We do not propose to defend this construal; we simply observe that the geometry does not refuse it.

The picture

The diagram below renders the situation at Six Mile Spring. The vertical axis is time, the horizontal axis is the distance toward Las Vegas. The solid black line is the porch’s worldline; the slanted dashed line is the worldline of the eastbound truck. The two grey lines are the respective hyperplanes of simultaneity through the event labelled Vegas, evening of 21 September. The relative tilt — exaggerated by a factor of for visibility — is the geometrical content of the disagreement.

import numpy as np
import matplotlib.pyplot as plt

fig, ax = plt.subplots()

# Spatial extent (km), time extent (arbitrary units)
x_max = 140
t_max = 10

# Worldlines
# Porch: vertical at x = 0
ax.plot([0, 0], [-t_max, t_max], color="#2b2118", lw=2, label="Porch worldline")
# Truck: tilted (exaggerated tilt for visibility)
truck_v = 0.35  # exaggerated
ax.plot([-t_max*truck_v*10, t_max*truck_v*10],
        [-t_max, t_max],
        color="#a0522d", lw=2, ls="--", label="Truck worldline")

# Las Vegas event
vegas_x, vegas_t = 110, 0
ax.plot(vegas_x, vegas_t, "o", color="#7a3b1f", markersize=9, zorder=5)
ax.annotate("Las Vegas\n21 Sept., evening", xy=(vegas_x, vegas_t),
            xytext=(vegas_x+3, vegas_t+0.7),
            fontsize=10, fontstyle="italic", color="#2b2118")

# Hyperplane of simultaneity for the porch (horizontal through Vegas event)
ax.plot([-x_max*0.15, x_max], [vegas_t, vegas_t],
        color="#7a8b6f", lw=1.4, alpha=0.9)
ax.text(x_max*0.05, vegas_t+0.18, "porch's 'now'",
        fontsize=9, color="#4f5e48", fontstyle="italic")

# Hyperplane of simultaneity for the truck: tilted by ~v in our exaggerated units
plane_slope = 0.025  # exaggerated
xs = np.array([-x_max*0.15, x_max])
ax.plot(xs, vegas_t + plane_slope*(xs - vegas_x),
        color="#b8794f", lw=1.4, alpha=0.9, ls="-.")
ax.text(x_max*0.05, vegas_t - 0.55, "truck's 'now'",
        fontsize=9, color="#7a3b1f", fontstyle="italic")

# Six Mile Spring marker
ax.plot(0, 0, "s", color="#2b2118", markersize=8, zorder=5)
ax.annotate("Six Mile Spring", xy=(0, 0), xytext=(-25, -1.2),
            fontsize=10, fontstyle="italic", color="#2b2118")

# Cosmetics
ax.set_xlim(-30, x_max+5)
ax.set_ylim(-t_max, t_max)
ax.set_xlabel("distance east (km)")
ax.set_ylabel("time (arb. units)")
ax.axhline(0, color="#d9b382", lw=0.5, alpha=0.5)
ax.axvline(0, color="#d9b382", lw=0.5, alpha=0.5)
ax.grid(alpha=0.15)
ax.legend(loc="lower right", framealpha=0.9, fontsize=9)
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
for spine in ax.spines.values():
    spine.set_color("#7a3b1f")
plt.tight_layout()
plt.show()

What follows, and what does not

It would be excessive to claim that Equation 2, by itself, decides the metaphysical question. The argument requires an additional premise — that reality is what is simultaneous with one’s present — and this premise has been variously qualified, denied, or replaced (see, among others, Stein (1968) and Savitt (2000)). The literature is patient and remains open.

What Equation 2 does establish, conclusively, is that any view of time which insists on a single objective now sweeping forward through the universe must explain why two observers, identically constituted, in demonstrably symmetrical situations, disagree about its location. The block universe — the view that all events of spacetime are equally real, and that the apparent flow of time is a feature of a particular kind of embedded perceiver and not of the world — is, at the very least, the view that demands the fewest auxiliary assumptions.

It has been remarked at the Institute, on more than one occasion, that the desert encourages the block-universe view. The night sky shows us the past; the rocks beneath show us a deeper past; the wind that has scoured them is, in the relevant geometrical sense, no less there than the boulder it is presently rounding. We have lived here long enough not to find this troubling.

References

The vertex function

Define the on-shell vertex by

with . The form factor is the anomalous moment. At one loop is infrared-finite even though is not — the IR divergences sit entirely in the charge renormalization piece.

The Feynman parameter integral

After the standard manipulations one arrives at

with , which collapses to

This recovers Equation 1.

A numerical sanity check

import numpy as np, matplotlib.pyplot as plt

alpha = 1/137.035999
# Coefficients C_n in a_e = sum_n C_n (alpha/pi)^n  (electron, QED only, schematic)
C = [0.5, -0.328478965, 1.181241, -1.9106, 9.16]
contribs = [C[n] * (alpha/np.pi)**(n+1) for n in range(len(C))]
cum = np.cumsum(contribs)

fig, ax = plt.subplots(figsize=(6.5, 3.8))
orders = np.arange(1, len(C)+1)
ax.semilogy(orders, np.abs(contribs), 'o-', color="#7a3b1f", label="|order n contribution|")
ax.semilogy(orders, np.abs(cum - cum[-1]) + 1e-20, 's--', color="#7a8b6f", label="|deviation from 5-loop sum|")
ax.set_xlabel("perturbative order $n$")
ax.set_ylabel(r"size of contribution to $a_e$")
ax.set_facecolor("#faf3e3")
fig.patch.set_facecolor("#faf3e3")
ax.grid(alpha=0.25)
ax.legend()
plt.tight_layout()
plt.show()

print(f"a_e (one-loop)  = {contribs[0]:.6e}")
print(f"a_e (five-loop) = {cum[-1]:.10e}")

a_e (one-loop)  = 1.161410e-03
a_e (five-loop) = 1.1596521777e-03

Why this still matters

The electron is the most precisely known prediction in physics. The agreement between five-loop QED and experiment is at the part-per-trillion level, and any disagreement in the next decimal place is news. The desert, unfortunately, contributes nothing to the running budget; we read about the new measurements like everyone else.