Modified Measure Formalism in Field Theory

• Modified Measure Formalism is a framework in field theory where standard metric-dependent integration measures are replaced by auxiliary scalar densities.
• It uses auxiliary fields to construct diffeomorphism-invariant measures that dynamically generate physical constants such as string or brane tension.
• The approach introduces novel gauge symmetries that protect against quantum corrections and establish unified boundary conditions for extended objects.

The modified measure formalism is a general approach in field theory and mathematical physics in which the standard metric-dependent integration measure in an action functional is replaced by a metric-independent or auxiliary-scalar-constructed density. Unlike the conventional use of $\sqrt{-\gamma} d^{p+1}\sigma$ (where $\gamma_{ab}$ is the worldvolume metric), the modified measure formalism introduces a new scalar density, allowing for fundamental parameters—such as string or brane tension—to be generated dynamically as constants of motion. This approach yields nontrivial boundary conditions and connections to higher derivative field theories, extended objects, and multifractal measures.

1. Canonical and Modified Measure Density

Traditionally, the invariant volume element for p-brane or string actions employs the determinant of the intrinsic metric: $\sqrt{-\gamma} \, d^{p+1}\sigma,$ with $\gamma_{ab}$ the worldvolume metric. The modified measure formalism replaces $\sqrt{-\gamma}$ by a new diffeomorphism-invariant scalar density constructed from auxiliary fields. For the string worldsheet, with coordinates $\sigma^a$ $(a=0,1)$ and two scalar fields $\varphi^i(\sigma),\, i=1,2$, the measure density takes the form

$$\Phi(\varphi) = \frac{1}{2} \epsilon_{ij} \epsilon^{ab} \partial_a \varphi^i \partial_b \varphi^j,$$

where $\epsilon^{ab}$ is the Levi-Civita symbol. This density transforms identically to $\sqrt{-\gamma}$ under diffeomorphisms, permitting its use as a substitute in any reparametrization-invariant action (Vulfs et al., 2018).

2. String and p-Brane Actions in Modified Measure Theory

A prototypical Polyakov-type string action within this framework is

$$S = -\int d^2 \sigma \, \Phi(\varphi) \left[\tfrac12 \gamma^{ab} \partial_a X^\mu \partial_b X^\nu g_{\mu\nu}(X) - \frac{\epsilon^{ab}}{2\sqrt{-\gamma}} F_{ab}(A)\right] + \int d^2 \sigma \, A_a j^a_{\rm charges},$$

where $F_{ab}(A) = \partial_a A_b - \partial_b A_a$ is an auxiliary Abelian gauge field strength, and $j^a$ is a localized current (e.g., for endpoint or junction charges in string models). For higher-dimensional branes, one constructs a similar measure, possibly with more auxiliary fields or forms. In the Galileon variant, the measure can be written as $\Phi(\chi) = \sqrt{-\gamma} \nabla^2 \chi$ using a scalar field $\chi$ (Vulfs et al., 2017).

3. Dynamical Generation of Tension and Equations of Motion

Variation of the action with respect to the metric, measure fields, and gauge fields leads to fundamental differences from standard formulations:

• Gauge Field Variation: Imposes

$$\epsilon^{ab} \partial_b \left(\frac{\Phi}{\sqrt{-\gamma}}\right) = 0 \implies \frac{\Phi}{\sqrt{-\gamma}} = T = \text{const.}$$

The tension $T$ is thus not an external parameter but an integration constant, determined dynamically.

• Measure Field Variation: Forces the combined kinetic-gauge Lagrangian to be a worldvolume constant,

$L = M = \text{const}.$

• Metric Variation: Recovers the Virasoro constraints (or generalizations for p-branes), as in standard approaches, due to the independence of $\Phi$ from the metric (Vulfs et al., 2018, Vulfs et al., 2017).

4. Dynamical Boundary Conditions: Neumann and Dirichlet

The presence of localized charges in the modified measure formalism yields discontinuities in the ratio $\Phi/\sqrt{-\gamma}$, corresponding physically to changes in tension. This has profound implications:

• Endpoint Charges: For a point-like charge $e$ at $\sigma_0$,

$$\partial_\sigma \Big(\frac{\Phi}{\sqrt{-\gamma}}\Big) = e \delta(\sigma - \sigma_0) \implies \frac{\Phi}{\sqrt{-\gamma}}(\sigma_0^+) - \frac{\Phi}{\sqrt{-\gamma}}(\sigma_0^-) = e,$$

and the motion equations enforce $\partial_\sigma X^\mu|_{\sigma_0}=0$, a Neumann boundary condition at each charge.

• String Intersections: When two strings are joined, a potential term induces Dirichlet conditions at the intersection: $X^\mu(\tau,\sigma_0) = Y^\mu(\tau,\sigma_0)$, ensuring the strings connect at the same spacetime point, with standard Neumann conditions at their endpoints. This resolves the nonlocality problem encountered in previous Lagrange-multiplier ("t Hooft junction") approaches (Vulfs et al., 2018).

5. Symmetry Principles and Quantum Protection

The modified measure formalism admits novel infinite-dimensional gauge-like symmetries, such as

$\varphi^i \to \varphi^i + f^i(L)$

where $L$ is the Lagrangian density. In the Galileon variant, a shift symmetry,

$\chi(\sigma) \to \chi(\sigma) + b_a \sigma^a + c,$

ensures that only second-derivative terms persist in the equations of motion, despite higher-derivative appearance at the action level. These symmetries protect the linearity of the action in the measure fields from quantum corrections, guaranteeing the stability of the mechanism that ties tension and boundary conditions to variational principles (Vulfs et al., 2017).

6. Generalizations and Applications

The modified measure approach applies broadly to p-branes, allowing the dynamical emergence of tension and nontrivial boundary conditions in any worldvolume dimension. One can select conformal measures for Weyl invariance, and, in cosmological settings, measures based on auxiliary fields can be engineered to describe dark sector interactions. At the level of mathematical measures, the formalism underlies certain multifractal structures: for instance, the construction of inhomogeneous multinomial measures (non-doubling under projection) leads to an extended multifractal formalism with distinct lower and upper scaling functions $b(q)$ and $B(q)$, and their Legendre transforms specifying the Hausdorff and packing dimension spectra, respectively (Shen, 2014).

7. Implications and Outlook

The modified measure formalism fundamentally alters the role of geometric and physical parameters in field theory and extended object dynamics. By elevating tension to a dynamical variable, this framework removes the need for ad hoc prescriptions of physical constants and provides a unified variational origin for Neumann and Dirichlet boundary conditions. It systematically resolves nonlocality issues in string junction models and is protected by novel symmetries. These features enable its adoption in string theory, extended object cosmology, and multifractal analysis, with further avenues including higher-brane intersections, Galileon field couplings, cosmological applications, and exploration of non-doubling measures in mathematical physics (Vulfs et al., 2018, Vulfs et al., 2017, Shen, 2014).