Gravity / The vacuum
On the lattice, the energy of empty space is a count.
Add up the zero-point energies of a quantum field to the Planck scale and empty space should weigh about 10−2 in Planck units; the expansion of the universe measures 1.1 × 10−123. In the lattice’s sum over geometries in real time the vacuum energy is a phase, and the phase is a count: π/4 for each mode, with its sign. The action has no volume term, and for the graviton and massless fields the count ignores the lattice’s size. Reading that as no cosmological constant is conditional; massive fields do feel the size, and no value of the vacuum energy is derived.
No volume term, the count, its scale invariance: provedRead as no cosmological constant: conditionalThe value of the vacuum energy: open
- Modes of the time interval
- Negative modes
- Zero-point energy times time
Live · exact, one oscillator
0.10π
- Negative
- 0modes with (nπ/T)² < ω²
- Phase
- 0π/4 per mode, with its sign, against the free particle
- Smooth
- −0.05πthe line −ωT/2: zero-point energy ℏω/2 times the time
One oscillator of angular frequency ω, followed for a time T. Its path integral is a product over the modes of the interval, and each mode contributes eiπ/4 or e−iπ/4 by the sign of its eigenvalue (nπ/T)² − ω². Each time ωT passes a multiple of π one eigenvalue turns negative and the phase steps by π/2; on average it falls like ωT/2, the zero-point energy times the time.
Cosmological-constant paper, Section 4 and Proposition 4.1
With JavaScript on, this figure follows one oscillator’s modes as the time grows: each time ωT passes a multiple of π, one eigenvalue turns negative and the phase steps by π/2, so on average it falls like ωT/2, the zero-point energy ℏω/2 times the time.
The estimate
10¹²⁰ times too much.
Zero-point energies up to a largest wave number K give K⁴/(16π²) for a massless field, in Planck units. With K at the Planck scale that is 6.3 × 10−3 of the Planck density; with the lattice’s own cutoff π/b, at b ≈ 1.92 Planck lengths, 4.5 × 10−2. The measured value is 1.1 × 10−123. If the lattice’s vacuum gravitated the way the estimate assumes, the lattice would make the problem worse, not better.
- K⁴/(16π²) at your cutoff
- Measured, Planck 2018
Live · the estimate of Section 1
1.0
- Cutoff
- 1.2×1028 eVas an energy
- Estimate
- 6.3×10−3Planck densities
- Against
- 5.6×10120 times the measured value
Where the numbers come from. The measured value is 3ΩΛ(H0tP)²/(8π) = 1.1 × 10−123 with ΩΛ = 0.6847 and H0 = 67.36 km s−1 Mpc−1, an energy scale of about 2.2 meV. A cutoff that keeps Lorentz invariance replaces K⁴ by the fourth power of each field’s mass; with the masses of the Standard Model the estimate is then more than fifty orders of magnitude too large instead of a hundred and twenty. The button that matches the measurement is computed here from K⁴/(16π²).
Cosmological-constant paper, Sections 1 and 6
With JavaScript on, choose the cutoff K and compare K⁴/(16π²) with the measured 1.1 × 10−123: 6.3 × 10−3 at the Planck scale, 4.5 × 10−2 at the lattice’s cutoff.
No volume term
The action cannot see the volume.
A cosmological constant would be a term ΛV in the action. The Regge action, a sum over triangles of area times deficit angle, vanishes on every flat configuration whatever its volume, and it is of degree one in the squared lengths while the volume is of degree two. So at every stationary point S = 2ΛV, and the flat lattice is a vacuum only when Λ = 0: in the path sum’s own weight, the native amplitude, the bare cosmological constant is zero, not free.
- S − ΛV along the flat scale ray
- A cost on cell volumes (a model)
Live · exact, Propositions 3.1 and 3.2
0.50
- Slope at 1
- −1.00S − 2ΛV, with S = 0 on the flat latticenot stationary: the flat lattice is not a vacuum
What the figure shows. Along the flat scale ray, every squared length multiplied by c, S stays 0 and S − ΛV = −Λc²V, which tilts the flat lattice toward larger or smaller sizes unless Λ = 0. The path-sum weight eiaS is exactly 1 all along the ray. A recognition cost J charged on each simplex’s volume ratio (a model) would hold the size like a spring, 24N⁴J(c²), zero with zero slope at c = 1, and never tilt it. Both are drawn per unit flat volume.
Cosmological-constant paper, Propositions 3.1 and 3.2; reading it as the ledger’s amplitude is the phase premise
1 · The vacuum is a count
π/4 per mode, with its sign.
As the regulator is removed, the phase of the one-loop vacuum of the path sum tends to π/4 times the number of positive modes minus the number of negative ones, once the zero modes are set aside. Read as the vacuum energy times the time extent, as for one oscillator (an inference), it is of order one per cell: the Planck-sized vacuum energy of the estimate. Whether it gravitates is a separate question.
Cosmological-constant paper, Proposition 4.1
2 · No response to the size
Rescale the lattice and the count stays.
The Regge action is homogeneous of degree one, so multiplying every squared length by c divides every eigenvalue of the graviton’s quadratic action by c and keeps every sign. The limiting phase of its one-loop vacuum, and a massless scalar’s on the flat lattice, is the same at every size. Reading that as “no cosmological constant from the graviton’s vacuum at one loop” is conditional: on the phase read as an energy, on the measure, and on the effective action of a path sum that does not yet have a massless graviton or a unitary evolution in record time.
Cosmological-constant paper, Propositions 4.2 and 4.3
3 · Massive fields
Mass breaks the scale invariance.
A field of mass m on a rescaled lattice is the same field with its mass multiplied by √c, so its vacuum responds to the size. The response is estimated at order (m/mP)² in Planck units, about 2 × 10−45 for the electron and 2 × 10−34 for the top quark (an inference: the coefficient is not computed). Nothing in the ledger as formulated cancels it.
Cosmological-constant paper, Proposition 4.3 and Section 4
4 · A phase per cell
It drops out if every history has the same number of cells.
A fixed vacuum phase per cell cancels from every normalized expectation when every history of the path sum has the same number of cells, and acts as a volume term when the number varies. So whether the lattice’s vacuum gravitates turns on a question about records: is their number the same in every history?
Cosmological-constant paper, Proposition 4.5
The expansion
A fixed lattice cannot expand.
For a homogeneous cell with scale factor a, the size of space, and lapse n, the proper time per rank of record time, the lattice value is κ = ¼ + n²/a². The programme reads proper time as a count of records, a fixed tick per rank, and the posting order as the causal order, so postings stay null at κ = 1. Together the two readings fix the scale factor: a fixed lattice cannot expand at all. So, the paper infers, a universe that expands grows by adding records, new cells at the fixed spacing, not longer edges.
- Null postings, κ = 1
- A fixed tick
- Your cell
Live · exact, Proposition 5.4
1.15
1.00
- κ
- 1.006¼ + n²/a²
- Cell
- admissible
- Postings
- null
The trilemma. Null postings and a fixed tick n₀ fix a² = 4n₀²/3, the filled point. A fixed tick alone keeps every simplex timelike in all its faces only while a < 2n₀, the open point, so from the null cell the scale factor can grow by less than √3. Null postings alone give n² = ¾a²: record time is conformal time.
Cosmological-constant paper, Proposition 5.4 and Figure 1; the step to “the universe grows by adding records” is an inference
Growing by records
A constant rate of new records is a cosmological constant.
For records of fixed size the scale factor grows as R1/3 for R records in a region, so the expansion rate is H = Γ/3, with Γ the rate of new records per record. A constant rate is de Sitter expansion, the steady exponential growth a cosmological constant drives, with Λ = Γ²/3. A whole-number count meets Friedmann’s equation, general relativity’s law for the expansion rate, exactly at every tick only by multiplying itself by a whole factor, a Planck-sized constant; a small one is reached in the limit of many records.
- Relative defect from Friedmann’s equation, each tick
Live · exact counts, Proposition 5.6
0.010
- Defect
- …at a million records, where … are added per tick
- Ticks
- …from one record to a billion
- ΛℓP²
- …the constant this rate stands for, (ln(1 + x))²/(12 ln 2)
The rule and the test. Each tick adds ⌈xR⌉ records. With the vacuum density (ln(1 + x))²/(24πGτ0²) that a constant rate needs, the relative defect H²/((8πG/3)ρ) − 1 at each tick is (ln(Rnext/R)/ln(1 + x))² − 1. Doubling meets Friedmann’s equation exactly, with ΛℓP² = ln 2/12; a small rate meets it only as the count grows, with a defect at most a multiple of one over the number of records created per tick. τ0² = 4 ln 2 Planck times squared is the tick of the identifications.
Cosmological-constant paper, Propositions 5.5 and 5.6 and Target 5.7
With JavaScript on, choose a rate x and watch a whole-number count meet Friedmann’s equation only as it grows: doubling each tick meets it exactly, with a Planck-sized constant ln 2/12.
The measured rate
5×10−61
new records per record per tick: the measured constant, ΓtP = 2.92 × 10−61, is one new record per record in about 3 × 1060 Planck times. Why it is so small is the question this reading leaves.
Cosmological-constant paper, Proposition 5.5 (comparison; the reading of de Sitter expansion as record creation is an inference)
Exact at every tick
ln 2/12
the least ΛℓP² a whole-number count can give if it meets Friedmann’s equation exactly at every tick with a vacuum alone: the count must grow by a whole factor, at least 2, each tick. The measured value is about 10120 times below it.
Cosmological-constant paper, Proposition 5.6
Power laws
w = 2/p − 1
A count R ∝ tp is a perfect fluid: p = 2 dust, p = 3/2 radiation, p = 3 a coasting universe, p = 1 a stiff fluid. At long wavelengths the quadratic part of the lattice’s homogeneous action is Friedmann’s, with G = 1/(8πs).
Cosmological-constant paper, Propositions 5.2 and 5.5
Open
How?
How the ledger adds records: a law whose count is the same in every history and meets Friedmann’s equation in the limit of many records, with the ledger’s own matter. The same count decides whether a vacuum phase per cell gravitates.
Cosmological-constant paper, Target 5.7 and Proposition 4.5
One number for two constants. In the ledger’s own units the measured vacuum energy is a single number, X = ρΛc²λrec⁴/(ℏc), which lies between 8.4 and 9.0 × 10−123 at tree level (a comparison). The same number fixes Newton’s constant: with H0 and ΩΛ measured, a derived X would give G = πXc⁵/(6(ln 2)²ℏH0²ΩΛ) to the 2% precision of H0²ΩΛ, under the identifications of the Newton’s constant paper. A derivation of the vacuum energy on the ledger would therefore also be a prediction of G. The one number that fixes G
Cosmological-constant paper, Section 6
No value of the vacuum energy is derived on this page or in the paper: the numbers it would have to meet are measurements, and none of them enters a proof. The premises: κ = 1, the phase premise, and the two readings behind the trilemma, that proper time counts records and that the posting order is the causal order.
Source: the cosmological-constant paper is The Cosmological Constant on the Recognition Lattice: Vacuum Energy, Record Creation and the Expansion of the Universe, version 1.2.
Words on this page, in plain terms: record · tick · Regge action · deficit angle · path sum · phase premise.
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