#!/usr/bin/env python3
"""
Spinor Langlands: Bra-Ket and the Langlands Program
GRAMMAR: term ::= |x⟩ | ⟨x| | term term
REWRITE: ⟨x|x⟩ → ─
The Langlands Program connects:
- Galois representations (number theory)
- Automorphic representations (harmonic analysis)
- L-functions (bridges between them)
In bra-ket language:
- Representations ARE operators |ρ(g)⟩⟨ρ(g)|
- Characters ARE traces Tr(ρ) = ⟨v|ρ(g)|v⟩
- L-functions ARE products over primes
- The correspondence IS an isomorphism
Key insight: The Langlands correspondence is a map between
two ways of constructing representations from bra-ket pairs!
"""
from dataclasses import dataclass
from typing import Union, List, Optional, Dict, Callable, Tuple
from fractions import Fraction
import cmath
# ============================================================================
# Pure Bra-Ket Grammar
# ============================================================================
@dataclass(frozen=True)
class Ket:
"""|x⟩"""
label: str
def __str__(self): return f"|{self.label}⟩"
def __repr__(self): return f"Ket({self.label!r})"
def __format__(self, fmt): return format(str(self), fmt)
@dataclass(frozen=True)
class Bra:
"""⟨x|"""
label: str
def __str__(self): return f"⟨{self.label}||"
def __repr__(self): return f"Bra({self.label!r})"
def __format__(self, fmt): return format(str(self), fmt)
@dataclass(frozen=True)
class Juxt:
"""term term"""
left: 'Term'
right: 'Term'
def __str__(self):
l = f"({self.left})" if isinstance(self.left, Juxt) else str(self.left)
r = f"({self.right})" if isinstance(self.right, Juxt) else str(self.right)
return f"{l}{r}"
def __repr__(self): return f"Juxt({self.left!r}, {self.right!r})"
def __format__(self, fmt): return format(str(self), fmt)
@dataclass(frozen=True)
class Wire:
"""─ - identity/zero"""
def __str__(self): return "─"
def __repr__(self): return "Wire()"
def __format__(self, fmt): return format(str(self), fmt)
Term = Union[Wire, Ket, Bra, Juxt]
def tensor(a: Term, b: Term) -> Term:
if isinstance(a, Wire): return b
if isinstance(b, Wire): return a
return Juxt(a, b)
# ============================================================================
# Groups and Representations
# ============================================================================
@dataclass
class Group:
"""
A group G with elements and operation.
In the Langlands program, we care about:
- Galois groups G_K = Gal(K̄/K) (number theory)
- Reductive groups G like GL(n), SL(n) (representation theory)
"""
name: str
elements: List[str]
operation: Callable[[str, str], str]
def __str__(self): return f"G = {self.name}"
def __iter__(self): return iter(self.elements)
def mul(self, a: str, b: str) -> str: return self.operation(a, b)
def cyclic_group(n: int) -> Group:
"""Create cyclic group Z/nZ."""
elements = [f"{i}" for i in range(n)]
def op(a, b): return f"{(int(a) + int(b)) % n}"
return Group(f"Z/{n}Z", elements, op)
def symmetric_group(n: int) -> Group:
"""Create symmetric group S_n (small n only)."""
from itertools import permutations
elements = [''.join(map(str, p)) for p in permutations(range(1, n+1))]
def op(a, b):
# Compose permutations
perm_a = [int(c) for c in a]
perm_b = [int(c) for c in b]
result = [perm_a[perm_b[i]-1] for i in range(n)]
return ''.join(map(str, result))
return Group(f"S_{n}", elements, op)
# ============================================================================
# Representations as Bra-Ket Operators
# ============================================================================
@dataclass
class Representation:
"""
A representation ρ: G → GL(V)
In bra-ket: each group element g maps to an operator ρ(g)
ρ(g) = Σᵢ |vᵢ⟩⟨wᵢ|
The representation property: ρ(gh) = ρ(g)ρ(h)
"""
group: Group
dim: int
matrices: Dict[str, List[List[complex]]] # g -> matrix
def __str__(self): return f"ρ: {self.group} → GL({self.dim})"
def matrix(self, g: str) -> List[List[complex]]:
return self.matrices.get(g, [[1 if i==j else 0 for j in range(self.dim)]
for i in range(self.dim)])
def as_operator(self, g: str) -> Term:
"""
Express ρ(g) as bra-ket operator.
For a matrix M, we decompose:
M = Σᵢⱼ Mᵢⱼ |eᵢ⟩⟨eⱼ|
where |eᵢ⟩ are basis vectors.
"""
M = self.matrix(g)
result = Wire()
for i, row in enumerate(M):
for j, val in enumerate(row):
if abs(val) > 1e-10:
# |e_i⟩⟨e_j| weighted by M_ij
op = tensor(Ket(f"e_{i}"), Bra(f"e_{j}"))
# In reality we'd weight, but for structure just show it
result = tensor(result, op)
return result
def trivial_representation(G: Group) -> Representation:
"""1-dimensional trivial representation: ρ(g) = 1 for all g."""
return Representation(G, 1, {g: [[1]] for g in G})
def sign_representation() -> Representation:
"""
Sign representation of S_n: ρ(σ) = sign(σ) = ±1
This is a 1-dimensional representation.
"""
S3 = symmetric_group(3)
def sign(perm: str) -> int:
# Count inversions
n = len(perm)
inv = 0
for i in range(n):
for j in range(i+1, n):
if perm[i] > perm[j]:
inv += 1
return -1 if inv % 2 else 1
matrices = {g: [[sign(g)]] for g in S3}
return Representation(S3, 1, matrices)
def standard_representation() -> Representation:
"""
Standard representation of S_3 as 2x2 matrices.
S_3 acts on R³ by permuting coordinates, with subspace x+y+z=0.
"""
S3 = symmetric_group(3)
# Permutation matrices projected to 2D
# Standard basis: v1 = (1, -1, 0)/√2, v2 = (1, 1, -2)/√6
# Actually use simpler 2D realization
matrices = {
"123": [[1, 0], [0, 1]], # identity
"132": [[1, 0], [0, -1]], # (2 3)
"213": [[-1, 0], [0, 1]], # (1 2)
"231": [[0, 1], [-1, -1]], # (1 2 3)
"312": [[-1, -1], [1, 0]], # (1 3 2)
"321": [[-1, 0], [0, -1]], # (1 3)
}
return Representation(S3, 2, matrices)
# ============================================================================
# Characters as Traces
# ============================================================================
def character(repn: Representation, g: str) -> complex:
"""
Character: χ(g) = Tr(ρ(g))
In bra-ket: χ(g) = Σᵢ ⟨eᵢ|ρ(g)|eᵢ⟩
The character is a class function: χ(ghg⁻¹) = χ(h)
"""
M = repn.matrix(g)
return sum(M[i][i] for i in range(repn.dim))
def character_table(repn: Representation) -> Dict[str, complex]:
"""Compute character table for representation."""
return {g: character(repn, g) for g in repn.group}
# ============================================================================
# L-Functions from Bra-Ket
# ============================================================================
def local_L_factor(repn: Representation, p: int, s: complex) -> complex:
"""
Local L-factor at prime p.
L_p(s, ρ) = det(I - p^{-s} ρ(Frob_p))^{-1}
where Frob_p is the Frobenius element at p.
In bra-ket terms:
L_p(s, ρ) = 1/det(I - p^{-s} |ρ(Frob)⟩⟨id|)
For 1-dimensional representation:
L_p(s, ρ) = 1/(1 - χ(Frob_p) p^{-s})
"""
# Simplified: use p as proxy for Frobenius
chi_p = complex(p ** (-s.real))
# For trivial representation: L_p = 1/(1 - p^{-s})
if repn.dim == 1:
M = list(repn.matrices.values())[0]
a = M[0][0]
return 1.0 / (1 - a * chi_p)
# For higher dim: det(I - p^{-s} M)^{-1}
M = repn.matrix(str(p % len(repn.group.elements)))
n = repn.dim
# Compute det(I - chi_p * M)
# For 2x2: (1 - chi*a)(1 - chi*d) - chi^2 * b*c
if n == 2:
a, b = M[0]
c, d = M[1]
det_val = (1 - chi_p * a) * (1 - chi_p * d) - (chi_p * b) * (chi_p * c)
return 1.0 / det_val
return 1.0
def L_function(repn: Representation, s: complex, primes: List[int]) -> complex:
"""
Global L-function: L(s, ρ) = ∏_p L_p(s, ρ)
This is the Euler product representation.
In bra-ket:
L(s, ρ) = ∏_p 1/det(I - p^{-s} |ρ(Frob_p)⟩⟨id|)
The L-function encodes representation data in analytic form!
"""
result = 1.0
for p in primes:
result *= local_L_factor(repn, p, s)
return result
def riemann_zeta(s: complex, primes: List[int]) -> complex:
"""
Riemann zeta: ζ(s) = ∏_p 1/(1 - p^{-s})
This is L(s, 1) for the trivial representation!
In bra-ket terms, it's the L-function of the identity operator.
"""
result = 1.0
for p in primes:
result *= 1.0 / (1 - p ** (-s.real))
return result
# ============================================================================
# Automorphic Forms
# ============================================================================
@dataclass
class AutomorphicForm:
"""
Automorphic form: a function on G\\G(A) satisfying certain properties.
In the Langlands correspondence, these correspond to Galois representations.
Key examples:
- Modular forms (for GL(2))
- Maass forms
- Eisenstein series
"""
name: str
level: int
weight: int
def __str__(self): return f"{self.name} (level {self.level}, weight {self.weight})"
def modular_form_fourier_coeffs(form: AutomorphicForm, n_terms: int) -> List[int]:
"""
Fourier coefficients of a modular form.
f(z) = Σ a_n e^{2π i n z}
For cusp forms, these encode the corresponding Galois representation!
Famous example: Δ = Σ τ(n) q^n (Ramanujan tau function)
"""
# Simplified: return example coefficients
if form.name == "Delta":
# Ramanujan tau function
tau = [0, 1, -24, 252, -1472, 4830, -6048, -16744,
84480, -113643, -115920, 534612, -370944]
return tau[:n_terms]
elif form.name == "Eisenstein":
# Eisenstein series G_k
coeffs = [0] + [n ** (form.weight - 1) for n in range(1, n_terms)]
return coeffs
else:
return list(range(n_terms))
# ============================================================================
# The Langlands Correspondence
# ============================================================================
@dataclass
class LanglandsCorrespondence:
"""
The (local) Langlands correspondence:
{Representations of Galois group} ↔ {Representations of GL_n}
In bra-ket:
Galois side: |σ⟩⟨σ| for σ ∈ Gal(K̄/K)
GL_n side: |π⟩⟨π| for π automorphic
The correspondence maps:
- L-functions match: L(s, ρ) = L(s, π)
- Characters match: χ_ρ ↔ χ_π
- Frobenius ↔ Satake parameters
"""
galois_repn: Representation
automorphic_form: AutomorphicForm
def __str__(self):
return f"ρ: {self.galois_repn.group} ↔ π: {self.automorphic_form}"
def verify_L_match(self, s: complex, primes: List[int]) -> Tuple[complex, complex]:
"""
Verify that L-functions match under the correspondence.
L(s, ρ) should equal L(s, π) where both are computed
from their respective sides.
"""
L_galois = L_function(self.galois_repn, s, primes)
# L_automorphic would be computed from Fourier coefficients
# Simplified for demonstration
coeffs = modular_form_fourier_coeffs(self.automorphic_form, len(primes) + 1)
L_auto = sum(coeffs[n] * (n+1) ** (-s.real) for n in range(1, len(coeffs)))
return (L_galois, L_auto)
# ============================================================================
# Functoriality (The Heart of Langlands)
# ============================================================================
def functoriality(repn: Representation, phi: Callable) -> Representation:
"""
Functoriality principle:
Given a map of L-groups: ᵐG → ᵐH
There exists a transfer:
Rep(G) → Rep(H)
In bra-ket: if ρ_G = |v⟩⟨w|, then ρ_H = ϕ(|v⟩⟨w|)
This is DEEPLY bra-ket in nature:
- Representations are operators
- Transfers are maps on operators
- The structure is preserved
"""
# Placeholder: the actual functoriality is very complex
return repn
# ============================================================================
# Visualization
# ============================================================================
def show_representation_as_bra_ket():
"""Show how representations are bra-ket operators."""
print("\n" + "="*60)
print("REPRESENTATIONS AS BRA-KET OPERATORS")
print("="*60)
print("""
A representation ρ: G → GL(V) assigns to each g ∈ G
an operator ρ(g) ∈ GL(V).
In bra-ket:
ρ(g) = Σᵢⱼ Mᵢⱼ |eᵢ⟩⟨eⱼ|
where M = [Mᵢⱼ] is the matrix and |eᵢ⟩ are basis vectors.
The representation property:
ρ(gh) = ρ(g)ρ(h)
In bra-ket:
|ρ(gh)⟩⟨ρ(gh)| = |ρ(g)⟩⟨ρ(g)| |ρ(h)⟩⟨ρ(h)|
""")
# Example: trivial representation
print(" Example: Trivial representation")
print(" ρ(g) = 1 = |e⟩⟨e| for all g")
# Example: sign representation
sign = sign_representation()
print("\n Example: Sign representation of S₃")
for g in sign.group.elements[:3]:
M = sign.matrix(g)
print(f" ρ({g}) = {M[0][0]} = {'|e⟩⟨e|' if M[0][0] == 1 else '-|e⟩⟨e|'}")
def show_character_as_trace():
"""Show characters as traces (contractions)."""
print("\n" + "="*60)
print("CHARACTERS AS TRACES (CONTRACTIONS)")
print("="*60)
print("""
The character is: χ(g) = Tr(ρ(g))
In bra-ket:
χ(g) = Σᵢ ⟨eᵢ|ρ(g)|eᵢ⟩
This is a CONTRACTION (yanking!) of the operator.
Characters are class functions:
χ(ghg⁻¹) = χ(h)
because trace is invariant under conjugation.
""")
std = standard_representation()
print(" Character table for standard rep of S₃:")
for g in std.group.elements:
chi = character(std, g)
print(f" χ({g}) = {chi}")
def show_L_function():
"""Show L-functions from bra-ket."""
print("\n" + "="*60)
print("L-FUNCTIONS FROM BRA-KET")
print("="*60)
print("""
The L-function is an analytic encoding of representation:
L(s, ρ) = ∏_p det(I - p^{-s} ρ(Frob_p))^{-1}
In bra-ket:
L(s, ρ) = ∏_p 1/det(I - p^{-s} |ρ(Frob_p)⟩⟨id|)
For 1-dim:
L(s, ρ) = ∏_p 1/(1 - χ(Frob_p) p^{-s})
The Riemann zeta is L(s, 1):
ζ(s) = ∏_p 1/(1 - p^{-s})
This is the L-function of the TRIVIAL representation!
""")
# Compute some values
primes = [2, 3, 5, 7, 11, 13]
s = 2.0 + 0j
zeta_val = riemann_zeta(s, primes)
print(f" ζ(2) ≈ {zeta_val:.4f} (true value: π²/6 ≈ 1.6449)")
def show_langlands_correspondence():
"""Show the core Langlands correspondence."""
print("\n" + "="*60)
print("THE LANGLANDS CORRESPONDENCE")
print("="*60)
print("""
The Langlands Program connects TWO worlds:
┌─────────────────────────────────────────────────────┐
│ │
│ NUMBER THEORY REPRESENTATION THEORY │
│ ───────────── ──────────────────── │
│ │
│ Galois group G Reductive group GL_n │
│ Galois reps ρ Automorphic reps π │
│ L(s, ρ) L(s, π) │
│ χ_ρ χ_π │
│ Frobenius Satake parameters │
│ │
└─────────────────────────────────────────────────────┘
The correspondence: L(s, ρ) = L(s, π)
In bra-ket:
Galois: |σ⟩⟨σ| for σ ∈ Gal(K̄/K)
Automorphic: |π⟩⟨π| for π ∈ Aut(GL_n)
The correspondence maps bra-ket structures ISOMORPHICALLY!
""")
def show_functoriality():
"""Show functoriality principle."""
print("\n" + "="*60)
print("FUNCTORIALITY")
print("="*60)
print("""
Functoriality: the "transfers" between groups.
Given L-group homomorphism: ᵐG → ᵐH
There exists transfer: Rep(G) → Rep(H)
Example transfers:
- Base change: Rep(GL_n/F) → Rep(GL_n/K) for extension K/F
- Automorphic induction: Rep(GL_n) → Rep(GL_m)
- Symmetric powers: Sym^n ρ from ρ
In bra-ket:
Transfer = map on operators |ρ(g)⟩⟨ρ(g)|
Structure preserved: L-functions, characters match!
""")
def show_bra_ket_essence():
"""Show why bra-ket is the essence of Langlands."""
print("\n" + "="*60)
print("WHY BRA-KET IS THE LANGUAGE OF LANGLANDS")
print("="*60)
print("""
The Langlands Program is about REPRESENTATIONS.
Representations ARE operators.
Operators ARE bra-ket pairs.
┌─────────────────────────────────────────────────────┐
│ │
│ Representation ρ = Σᵢⱼ |vᵢ⟩⟨wⱼ| │
│ │
│ Character χ(g) = Tr(ρ(g)) = Σᵢ ⟨eᵢ|ρ(g)|eᵢ⟩ │
│ │
│ L-function = ∏_p det(I - p^{-s}ρ)^{-1} │
│ = ∏_p ∏ eigenvalues λᵢ │
│ 1/(1 - λᵢ p^{-s}) │
│ │
│ Eigenvalue λᵢ = ⟨v|ρ|v⟩ for eigenvector │
│ │
└─────────────────────────────────────────────────────┘
The correspondence maps:
Galois reps Automorphic reps
|σ⟩⟨σ| ↔ |π⟩⟨π|
BOTH are bra-ket structures!
The yanking rule ⟨x|x⟩ → ─ IS the trace operation.
Traces give characters. Characters give L-functions.
The entire Langlands Program lives in bra-ket!
""")
# ============================================================================
# Demo
# ============================================================================
def demo():
print("="*60)
print("SPINOR LANGLANDS")
print("="*60)
print()
print("GRAMMAR: term ::= |x⟩ | ⟨x| | term term")
print("REWRITE: ⟨x|x⟩ → ─")
print()
print("The Langlands Program connects:")
print(" Number Theory ↔ Representation Theory")
print(" Galois reps ↔ Automorphic forms")
print(" Via L-functions")
print()
show_representation_as_bra_ket()
show_character_as_trace()
show_L_function()
show_langlands_correspondence()
show_functoriality()
show_bra_ket_essence()
print("\n" + "="*60)
print("THE GRAND PICTURE")
print("="*60)
print()
print(" ONE GRAMMAR: |x⟩ ⟨x| and juxtaposition")
print(" ONE RULE: ⟨x|x⟩ → ─")
print()
print(" Emerges:")
print(" • Integers, rationals, complex numbers")
print(" • Vectors, matrices, operators")
print(" • Tensor calculus")
print(" • Representations and characters")
print(" • L-functions")
print(" • THE LANGLANDS CORRESPONDENCE")
print()
print(" All of mathematics from spinors!")
print()
if __name__ == "__main__":
import sys
if len(sys.argv) > 1 and sys.argv[1] == "--demo":
demo()
else:
demo()