Physical setups: fields, monomials, and reproducible files

A physical setup records the field content and conventions needed to ask a definite local-operator question. In this edition, the question is: how many independent complex, derivative-free scalars under connected Lorentz/spin and gauge transformations have the specified field powers?

Open the local-operator workbench, or download the physical-setup JSON schema. The counting method explains the exact character calculation. The format implements a bounded part of the broader physical-setup blueprint; a complete theory also requires choices beyond these data.

What the fields mean

The spacetime entry fixes four-dimensional Lorentzian signature. A field is a scalar, a left-handed Weyl spinor, or a right-handed Weyl spinor. Parity and time reversal are additional choices, not imposed here. Its gauge representation is specified by $SU(3)$ Dynkin labels $[p,q]$, weak $SU(2)$ Dynkin label $n$, and integer circle charge $Q$. The circle parameter has period $2\pi$, so its action is $e^{iQ\theta}$. Standard Model examples use $Q=6Y$.

The flavors entry counts independent copies with the same spin and gauge representation. Flavor remains an ungauged multiplicity space. Identical scalar occurrences enter symmetric powers; identical Weyl occurrences enter exterior powers of the entire spin, gauge, and flavor space. This prescription implements statistics before extracting invariants; see Lehman and Martin, §§2–4 and Appendix A and the field-statistics guide.

An id is a stable machine identifier used by the powers object. A name is a display label, so it can contain a dagger or a Greek letter. Distinct rows represent distinct field species even when their representations and display names agree. Renaming a field’s ID requires updating its power key.

A complete example

This setup requests the three-flavor Weinberg sector:

{
  "schema": "groups.qft.org/physical-setup/1",
  "name": "Weinberg operator — three lepton flavors",
  "spacetime": { "dimension": 4, "signature": "Lorentzian" },
  "gauge": {
    "kind": "su3-su2-u1",
    "quotientOrder": 6,
    "u1Normalization": "integer-charge-period-2pi"
  },
  "fields": [
    {
      "id": "L", "name": "L", "kind": "left-weyl",
      "su3": [0, 0], "su2": 1, "charge": -3, "flavors": 3
    },
    {
      "id": "H", "name": "H", "kind": "scalar",
      "su3": [0, 0], "su2": 1, "charge": 3, "flavors": 1
    }
  ],
  "operator": {
    "powers": { "L": 2, "H": 2 },
    "derivatives": 0,
    "equivalence": "algebraic",
    "coefficientField": "C"
  }
}

The requested monomial has canonical dimension $2(3/2)+2=5$. The calculation returns six complex invariants, corresponding to a symmetric three-flavor tensor. The Weinberg guide derives the contraction and its flavor symmetry. The Yukawa and four-fermion guide explains why superficially similar flavor problems instead give unrestricted matrices or more involved permutation relations.

Global admissibility comes first

The selected gauge group is

$$G_d=\frac{SU(3)\times SU(2)\times U(1)}{\Gamma_d}, \qquad \Gamma_d=\langle\gamma^{6/d}\rangle, \qquad d\in\{1,2,3,6\},$$

with the explicitly embedded generator

$$\gamma=\left(e^{2\pi i/3}\mathbf1_3,-\mathbf1_2,e^{2\pi i/6}\right).$$

This is the integer-charge convention used by Tong, §§1 and 3. Substituting the center characters gives the descent test for a field:

$$2(p+2q)+3n+Q\equiv0\pmod d.$$

Every declared field must satisfy this condition, including a field with exponent zero in the selected monomial. A structurally valid setup may fail descent. In that case the workbench identifies the incompatible fields and withholds the operator count. The setup can still be saved or transferred to the product gauge-group tool for diagnosis.

Once all fields descend, a fixed local polynomial has the same invariant count under each compatible central quotient: the quotient acts through the same field representations. This statement concerns local invariants; it does not determine the spectrum of line operators or other global physical choices.

Complex coefficients and explicit conjugation

Add a conjugate row when the polynomial needs a conjugate field. Conjugation exchanges $[p,q]$ with $[q,p]$, reverses $Q$, exchanges left and right Weyl chirality, and preserves the flavor multiplicity. The weak representation has the same Dynkin label. These representation facts underlie the two-component conventions reviewed by Dreiner, Haber, and Martin, §§2–3.

The added row remains an explicit variable. No Hermiticity condition, reality constraint, or identification under flavor rotations is inferred. For example, the single invariant in $H^2(H^\dagger)^2$ is a one-dimensional space over $\mathbb C$; a Hermitian scalar potential adds a condition on its coefficient.

Supported bounds and rejected meanings

Input	Contract
Declared fields	At most 16; distinct IDs
Field IDs	ASCII letter followed by letters, digits, or underscores; at most 24 characters; reserved keys excluded
Display names	At most 48 characters per field; at most 100 for the setup; single-line text
Color labels	Nonnegative integers with $p+q\leq2$
Weak label	$n=0,1,2$
Circle charge	Integer $-24\leq Q\leq24$
Flavor multiplicity	1, 2, or 3
Monomial	At most six field occurrences and canonical dimension at most six
Derivatives	Exactly zero
Equivalence and coefficients	Algebraic identities over $\mathbb C$

Missing powers of declared fields normalize to zero. Unknown fields in the powers object are rejected. All top-level entries and field attributes are required. Unknown semantic properties, sparse arrays, numeric strings, unsupported signature, derivatives, or equivalence prescriptions produce an error. An empty field list with empty powers denotes the constant invariant.

Algebraic counting includes graded commutation and relations among contractions. It does not impose equations of motion or integration by parts. Those require additional data and machinery; Henning, Lu, Melia, and Murayama, §2 explains their role in full EFT operator enumeration. Computational resource bounds are reported separately from these mathematical input bounds.

Save, import, and recompute

The workbench accepts either a raw setup as above or a groups.qft.org/physical-calculation/1 envelope. The envelope includes the canonical setup, result, source and convention metadata, application and dataset versions, and both the local-operator and gauge-descent algorithm versions. Exact large integers remain strings in JSON.

Import reads at most 4 MiB of UTF-8 JSON. It validates the setup and recomputes with the current algorithms. Cached answers, displayed formulas, and saved source text never enter the computation. A changed or missing saved algorithm version is reported explicitly; a matching version still triggers recomputation. Retain the original export when an earlier result is evidence for published work.

Worked contractions appear only for an exact matching published example. Editing its field data, names, global form, or powers removes that association. The general engine returns invariant counts and checks; these examples do not turn it into a generator of normalized contraction bases.