Expressions¶

Chacana parses tensor calculus expressions using a PEG grammar. This page covers the supported syntax.

Tensor expressions¶

A tensor with indices uses curly braces:

R{^a _b _c _d}     # Riemann tensor: 1 contra, 3 covariant
T{^a _b}            # Mixed rank-2 tensor
g{_a _b}            # Metric tensor (covariant)

^ marks a contravariant (upper) index
_ marks a covariant (lower) index

Bare identifiers without braces are also valid:

A                   # Bare tensor (no explicit indices)

Arithmetic¶

A{^a} + B{^a}      # Sum (free indices must match)
A{^a} - B{^a}      # Subtraction
A{_a} * B{^a}      # Product with contraction
3 * T{^a _b}       # Scalar multiplication
3.14                # Scalar literal
(A + B) * C        # Parenthesized sub-expression

Wedge product¶

A ^ B               # Exterior (wedge) product
d(A ^ B)            # Exterior derivative of a wedge product

Index derivatives¶

T{^a _b ;c}         # Covariant derivative (semicolon)
T{^a _b ,c}         # Partial derivative (comma)
T{;a ;b}            # Double covariant derivative

Symmetrization¶

T{_( a b _)}        # Symmetrize over a, b
T{_[ a b _]}        # Anti-symmetrize over a, b
T{^( a b ^)}        # Contravariant symmetrization

Nested symmetrization

Nested symmetrization like T{_( _( a b _) _)} is rejected by the parser as it is not supported in the Chacana micro-syntax.

Functional operators¶

d(omega)            # Exterior derivative
L(X, T{^a _b})     # Lie derivative
Tr(T{^a _b})       # Trace
det(g{_a _b})      # Determinant
inv(M)              # Inverse
star(omega)         # Hodge star (requires active_metric)
i(X, omega)         # Interior product

Perturbations and commutators¶

@2(g{_a _b})        # Second-order perturbation
[A, B]              # Commutator

Greek indices¶

Greek letters are supported and automatically detected as IndexType.GREEK:

T{^α _β}           # Greek indices
g{_μ _ν}           # Metric with Greek indices

Unicode normalization¶

All input is NFC-normalized before parsing. Decomposed characters (e.g., α + combining acute) are composed to their precomposed form.

Characters from non-Latin/Greek Unicode blocks (Cyrillic, Arabic, CJK, etc.) are rejected to prevent visual spoofing.