Interactive Digital Reproduction

The Laws of Symmetry

A Complete Theoretical Framework for the Conservation, Projection, and Origin of Symmetry in Physical Systems Across All Dimensions
Subhadip Mitra
Kshitij 2006
Indian Institute of Technology Kharagpur
2006
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About This Reproduction

This is an interactive digital reproduction of a theoretical physics paper originally presented at Kshitij 2006, the annual techno-management festival of the Indian Institute of Technology Kharagpur.

The original work, written by Subhadip Mitra as an undergraduate student, proposed a bold new framework for understanding symmetry in physics - arguing that symmetry is never truly broken, only hidden through dimensional projection.

This HTML edition preserves the complete original text while adding:

  • 🎨 Modern typesetting with beautifully rendered equations
  • 📊 Interactive animated diagrams that bring the concepts to life
  • 🧭 Navigation sidebar for easy exploration of all 23 chapters
  • 📱 Responsive design for reading on any device

The theoretical framework, predictions, and philosophical arguments remain exactly as written in 2006 - a time capsule of ambitious undergraduate physics theorizing, now presented for a new generation of readers.

Original Venue Kshitij 2006, IIT Kharagpur
Original Author Subhadip Mitra
Digital Reproduction December 2024
"The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect."
- Paul Valéry
"Symmetry is what we see at a glance."
- Blaise Pascal

Abstract

This treatise proposes a complete axiomatic framework - the Laws of Symmetry - governing the behaviour of symmetry in physical systems. Standing alongside Newton's Laws of Motion and the Laws of Thermodynamics, these laws establish symmetry as a conserved quantity that can be neither created nor destroyed, only redistributed across subsystems, dimensional subspaces, and correlational structures.

The central thesis is that symmetry is never truly broken - only hidden. What we observe as spontaneous symmetry breaking is reinterpreted as dimensional projection: the symmetry persists in the full higher-dimensional space but becomes invisible to observers confined to lower-dimensional subspaces.

The framework comprises five laws: The Zeroth Law establishes symmetry equivalence as transitive. The First Law asserts absolute conservation. The Second Law introduces dimensional projection. The Third Law addresses cosmological origins. The Fourth Law governs dynamics of symmetry transfer.

We develop extensive mathematical machinery including symmetry algebras, dimensional projection operators, and information-theoretic measures. Connections are established to quantum entanglement, thermodynamics, and the CPT theorem. The framework yields predictions for Kibble-Zurek scaling, LHC signatures, gravitational wave backgrounds, and precision CPT tests.

Keywords: symmetry; conservation laws; dimensional projection; spontaneous symmetry breaking; higher dimensions; Noether theorem; gauge theory; entropy; quantum entanglement; cosmology; Big Bang; CPT theorem
PART I
Foundations
CHAPTER 1

Introduction

1.1 The Primacy of Symmetry

What is the most fundamental concept in physics? One might answer: energy, or force, or spacetime. But I shall argue that the answer is symmetry. Every great unification in the history of physics has been achieved by recognising a deeper symmetry underlying apparently disparate phenomena. Every fundamental law of nature is an expression of symmetry. Every conservation law is a consequence of symmetry.

Consider the arc of physics from Newton to the present day. Newton unified terrestrial and celestial mechanics under laws invariant to spatial translations and rotations. Maxwell unified electricity and magnetism under equations invariant to Lorentz transformations, paving the way for Einstein. Einstein elevated symmetry to a principle - special relativity is the physics compatible with Lorentz symmetry; general relativity is the physics compatible with general coordinate invariance. The Standard Model is entirely determined by the gauge group SU(3) × SU(2) × U(1) and the requirement of local symmetry.

1.2 The Central Thesis

This treatise proposes a radical reframing of our understanding of symmetry, encapsulated in a single bold claim:

CENTRAL THESIS Symmetry is never broken, only hidden.

What we perceive as asymmetry is not the destruction of symmetry but its redistribution. When a ferromagnet 'breaks' rotational symmetry by choosing a preferred direction, the symmetry does not vanish - it is encoded in the degeneracy of ground states, in the Goldstone modes, in correlations that extend across the system.

More profoundly, I propose that our very perception of asymmetry arises from our confinement to a lower-dimensional subspace of a higher-dimensional reality. Just as the inhabitants of Flatland, confined to two dimensions, would perceive a passing sphere as a circle that mysteriously changes size, we, confined to 3+1 dimensions, perceive the shadows of higher-dimensional symmetric objects as asymmetric phenomena.

1.3 The Five Laws

I propose five Laws of Symmetry that govern the behaviour of symmetry in physical systems:

ZEROTH LAW (Equivalence) Symmetry equivalence is a transitive relation.
FIRST LAW (Conservation) The total symmetry of an isolated system is conserved.
SECOND LAW (Projection) Dimensional projection reduces apparent symmetry.
THIRD LAW (Origin) The Big Bang created symmetry with spacetime.
FOURTH LAW (Dynamics) Symmetry transfer rate is governed by energy and dimensional distance.
CHAPTER 2

Historical Development

2.1 From Geometry to Physics

The concept of symmetry is as old as human aesthetic sense. The bilateral symmetry of the human body, the rotational symmetry of flowers, the translational symmetry of crystals - these patterns have fascinated humanity since prehistory. The ancient Greeks elevated symmetry to a philosophical principle, with Plato associating the five regular polyhedra with the fundamental elements of nature.

But symmetry as a mathematical concept emerged only in the 19th century with the development of group theory. Évariste Galois, in his tragically brief life, discovered that the solvability of polynomial equations depends on the symmetry group of their roots. Sophus Lie extended these ideas to continuous transformations, founding the theory of Lie groups and Lie algebras central to modern physics.

2.2 Noether's Revolution

The profound connection between symmetry and conservation was established by Emmy Noether in her 1918 theorem [1]: every continuous symmetry of a physical system's action corresponds to a conserved quantity. Time translation symmetry implies energy conservation; spatial translation implies momentum conservation; rotation implies angular momentum conservation.

2.3 Spontaneous Symmetry Breaking

The concept of spontaneous symmetry breaking emerged from condensed matter physics and was imported to particle physics by Nambu [2], Goldstone [3], and others. Goldstone's theorem established that spontaneous breaking of a continuous symmetry produces massless particles (Goldstone bosons). The Higgs mechanism [4] showed how gauge symmetry breaking can give mass to gauge bosons.

The standard narrative treats symmetry breaking as genuine destruction. This treatise challenges that narrative: the symmetry is not destroyed but hidden.

2.4 Higher Dimensions

Kaluza [7] and Klein [8] showed in the 1920s that five-dimensional general relativity, with the fifth dimension compactified to a small circle, reproduces four-dimensional gravity plus electromagnetism. String theory [9] extends this to ten or eleven dimensions. These developments provide the physical context for our Second Law.

CHAPTER 3

Mathematical Preliminaries

3.1 Group Theory Foundations

Definition 1 (Group). A group (G, ·) is a set G equipped with a binary operation satisfying: (i) Associativity; (ii) Identity: ∃e with ea = ae = a; (iii) Inverse: ∀a ∃a⁻¹ with aa⁻¹ = e.
Definition 2 (Symmetry Group). The symmetry group of a physical system S with action functional S[φ] is G = {g : S[gφ] = S[φ] for all field configurations φ}.
Definition 3 (Symmetry Magnitude). For a Lie group G, the symmetry magnitude is Σ(G) = dim(G). For finite groups, Σ(G) = log|G|.

3.2 Lie Groups and Algebras

Definition 4 (Lie Algebra). A Lie algebra 𝔤 is a vector space with bracket [·,·]: 𝔤 × 𝔤 → 𝔤 satisfying antisymmetry and the Jacobi identity.
Example 1 (Common Symmetry Groups). The rotation group SO(3) has dimension 3. The Lorentz group SO(3,1) has dimension 6. The Standard Model gauge group SU(3) × SU(2) × U(1) has dimension 12.

3.3 Dimensional Projection

Definition 7 (Dimensional Projection). Let M be an N-dimensional manifold with projection π: M → B onto an n-dimensional submanifold B. For any field φ on M, the projected field is φ̃ = ∫F φ dμF.
Lemma 1 (Dimensional Symmetry Decomposition). For any projection: dim(G) = dim(H) + dim(G/H).
Proof. This is immediate from dim(G/H) = dim(G) − dim(H) for homogeneous spaces. The symmetry dimensions that 'disappear' under projection are precisely those acting along fibers.
PART II
The Five Laws of Symmetry
CHAPTER 4

The Zeroth Law - Symmetry Equivalence

ZEROTH LAW OF SYMMETRY If system A is symmetric with system B under transformations of group G, and B is symmetric with C under G, then A is symmetric with C under G. Symmetry equivalence is an equivalence relation.
Definition 9 (G-Equivalence). Two configurations φ₁ and φ₂ are G-equivalent, written φ₁ ~G φ₂, if there exists g ∈ G such that gφ₁ = φ₂.
Theorem 1 (Zeroth Law - Formal). The relation ~G is an equivalence relation: reflexive, symmetric, and transitive.
Proof. Reflexivity: eφ = φ. Symmetry: if gφ₁ = φ₂, then g⁻¹φ₂ = φ₁. Transitivity: if g₁φ₁ = φ₂ and g₂φ₂ = φ₃, then (g₂g₁)φ₁ = φ₃.
CHAPTER 5

The First Law - Conservation of Symmetry

FIRST LAW OF SYMMETRY The total symmetry of an isolated system is conserved. Symmetry cannot be created or destroyed, only redistributed among subsystems, transferred between dimensional subspaces, or exchanged with external systems.

5.1 Mathematical Formulation

Definition 10 (Symmetry Content). The symmetry content decomposes as:
Σtotal = Σmanifest + Σhidden + Σcorrelated
Theorem 2 (Conservation of Symmetry). For an isolated system: dΣtotal/dt = 0. If observed symmetry decreases, hidden symmetry increases by the same amount.
FIGURE 1: Conservation of Symmetry Under Apparent Breaking
TOTAL SYMMETRY G (conserved)
MANIFEST
H ⊂ G
+
HIDDEN
G/H
dim(G) = dim(H) + dim(G/H) = constant
When symmetry appears to break from G to H, the 'lost' symmetry G/H persists in hidden form - encoded in Goldstone modes, vacuum degeneracy, or extra dimensions.
Example 2 (Ferromagnetism). SO(3) breaks to SO(2). The coset SO(3)/SO(2) ≅ S² parameterises magnetisation directions. Total: 3 = 1 + 2. ✓
CHAPTER 6

The Second Law - Dimensional Projection

SECOND LAW OF SYMMETRY Any observation confined to n < N dimensions perceives reduced symmetry. The apparent loss equals symmetry 'projected out' into unobserved dimensions. What appears as symmetry breaking is dimensional projection.

6.1 The Flatland Principle

Consider Abbott's Flatland: a 2D world. A 3D sphere passing through Flatland appears as a circle that grows, reaches maximum size, and shrinks to a point. The Flatlanders observe change and asymmetry; from the 3D perspective, the sphere is unchanging and perfectly symmetric.

FIGURE 2: The Flatland Principle - Dimensional Projection Creates Apparent Asymmetry
3D REALITY
3D Observer sees: Unchanging sphere with perfect SO(3) symmetry
2D OBSERVATION
2D Flatlander sees: Circle that mysteriously grows and shrinks
A sphere passing through a plane appears as a changing circle. We are Flatlanders in 3+1 dimensions.
Theorem 4 (Symmetry Projection). Let M be N-dimensional with symmetry G, and π: M → B projection onto n dimensions. The observed symmetry H = {g ∈ G : π ∘ g = g ∘ π}. Hidden symmetry K = G/H.
Proposition 2 (Kaluza-Klein). In M⁵ = M⁴ × S¹, the 5D Poincaré ISO(4,1) projects to ISO(3,1) × U(1). The photon is the Goldstone boson of 'broken' fifth translation.
CHAPTER 7

The Third Law - The Origin of Symmetry

THIRD LAW OF SYMMETRY The Big Bang represents the creation of symmetry through the creation of spacetime. Prior to this, symmetry was undefined. All subsequent asymmetry arises from dimensional evolution.
Definition 11 (Cosmological Symmetry Function). Let σ(t) = dim(G(t))/dim(G₀) ≤ 1, where G(t) is manifest symmetry at time t.
Theorem 5 (Cosmological Monotonicity). For fixed-dimension observers: dσ/dt ≤ 0. Manifest symmetry never spontaneously increases.
FIGURE 3: Cosmological Evolution of Symmetry
Symmetry
Time →
Big Bang Planck GUT EW Now
Total Σ (constant)
Manifest Σ
Hidden Σ
The history of the universe is a history of symmetry redistribution. Total symmetry is conserved while manifest symmetry decreases and hidden symmetry increases.
CHAPTER 8

The Fourth Law - Dynamics of Symmetry Transfer

FOURTH LAW OF SYMMETRY The rate of symmetry transfer between manifest and hidden sectors is proportional to available energy E and inversely proportional to dimensional distance D: dΣtransfer/dt ∝ E/D
Theorem 6 (Energy Threshold). Symmetry transfer becomes significant when E ≳ ℏc/Rcompact.
FIGURE 4: The Dimensional Hierarchy
N-DIMENSIONAL BULK
Full symmetry G
↓
COMPACT DIMENSIONS
Hidden symmetry G/H
↓
OUR 3+1D BRANE
Observed symmetry H
Our 3+1D brane is embedded in higher-dimensional bulk. Hidden symmetry resides in compact dimensions, accessible only at high energies.
PART III
Temporal Symmetry and the Arrow of Time
CHAPTER 9

Time as Emergent from Symmetry

The fundamental laws of physics are time-symmetric: for every process allowed by the laws, the time-reversed process is also allowed. Yet we experience profound time asymmetry. Our framework offers a new perspective: the arrow of time emerges from symmetry redistribution.

Conjecture 1 (Time as Broken Symmetry). The apparent flow of time is spontaneous symmetry breaking. In the full higher-dimensional description, past and future are symmetric.
Theorem 7 (Symmetry-Entropy Duality). Total entropy decomposes as: Stotal = Sthermal + Ssym + Scorrelations
CHAPTER 10

The Symmetry Arrow of Time

Definition 14 (Symmetry Arrow). The symmetry arrow points in the direction of decreasing manifest symmetry: dΣmanifest/dt ≤ 0
FIGURE 5: The Two Arrows of Time
Entropy
→
Manifest Σ
→
Hidden Σ
→
Big Bang Far Future
Conservation: Σtotal = Σmanifest + Σhidden = constant
Both arrows originate from the Big Bang's high symmetry, low entropy state.
CHAPTER 11

Unification of Temporal Arrows

Conjecture 2 (Unified Arrow). All arrows of time - thermodynamic, cosmological, psychological, quantum, radiative - derive from the symmetry arrow.

Thermodynamic: Entropy increase IS symmetry hiding.

Cosmological: Expansion correlates with symmetry breaking epochs.

Psychological: Memory requires records, requires entropy increase, requires symmetry hiding.

Quantum: Wave function collapse is symmetry projection.

PART IV
Information-Theoretic Formulation
CHAPTER 12

Symmetry as Information

Theorem 9 (Symmetry-Information). Manifest symmetry is inversely related to symmetry information: Σmanifest = Σtotal − Isym/kB ln 2
Proposition 4. Hidden symmetry equals information loss under dimensional projection.
CHAPTER 13

Quantum Entanglement and Hidden Symmetry

Conjecture 3 (Entanglement-Symmetry Correspondence). Sentanglement = kB Σhidden ln 2. Hidden symmetry manifests as entanglement with the hidden sector.
Theorem 10. If |Ψ⟩ is G-symmetric and projection breaks G to H, then Sentanglement ≥ kB ln|G/H|.
Example 3 (Schrödinger's Cat). The state |alive⟩ + |dead⟩ has alive↔dead exchange symmetry. Decoherence hides this in environmental correlations. The total system retains symmetry.
CHAPTER 14

The Holographic Perspective

Conjecture 4 (Holographic Symmetry). The holographic principle is symmetry projection. Bulk symmetry in N dimensions projects to boundary symmetry in (N−1) dimensions.
Proposition 6 (AdS/CFT as Second Law). The AdS/CFT correspondence exemplifies the Second Law. The 'missing' radial dimension encodes symmetry appearing as conformal invariance on the boundary.
PART V
Applications and Examples
CHAPTER 15

Applications to Particle Physics

15.1 Electroweak Symmetry

Proposition 7 (Electroweak as Projection). SU(2)×U(1) → U(1)EM is dimensional projection. The Higgs field parameterises the coset. Hidden symmetry encoded in vacuum structure and massive gauge bosons.

15.2 Grand Unification

Proposition 8 (GUT as Symmetry Restoration). At E ≫ 10¹⁶ GeV, hidden GUT symmetry becomes manifest. Coupling unification is required by the restored symmetry.

15.3 Supersymmetry

Conjecture 5. Supersymmetry is exact in full superspace. The apparent breaking to ordinary spacetime is dimensional projection.
CHAPTER 16

Applications to Cosmology

Conjecture 6 (Dimensional Baryogenesis). Matter-antimatter asymmetry is a projection effect. In full dimensions, perfect symmetry exists. Our brane received unequal projection.
Proposition 9 (Inflation as Redistribution). Inflation is the period of most rapid symmetry redistribution. Quantum fluctuations seed structure because perfect symmetry is impossible.
CHAPTER 17

Condensed Matter Applications

17.1 Phase Transitions

Ferromagnetism: SO(3) → SO(2). Hidden symmetry in domain structure and magnons. Total: 3 = 1 + 2. ✓

Superconductivity: U(1) hidden in condensate phase. All phases physically equivalent.

Crystallisation: Continuous translation → discrete. Hidden symmetry is coset R³/Z³.

Proposition 10 (Defects as Boundaries). Topological defects are regions where symmetry projection is singular - 'seams' between domains of different projection.
CHAPTER 18

Worked Examples

18.1 Kaluza-Klein Reduction

Setup

M⁵ = M⁴ × S¹ with coordinates xM = (xμ, x⁵), where x⁵ ∈ [0, 2πR].

Symmetry Accounting

Full symmetry: ISO(4,1), dim = 15

After projection: ISO(3,1) × U(1), dim = 11

Hidden: 15 − 11 = 4 dimensions

x⁵ → x⁵ + ε(xμ)  ⟹  Aμ → Aμ + ∂με

Fifth dimension translation becomes U(1) gauge transformation!

18.2 Ferromagnetic Phase Transition

High T: Manifest SO(3), dim = 3. Hidden: 0.

Low T: Manifest SO(2), dim = 1. Hidden: S², dim = 2.

Total = 1 + 2 = 3 = dim(SO(3)) ✓
PART VI
Experimental Predictions
CHAPTER 19

Testable Consequences

19.1 Kibble-Zurek Mechanism

Prediction 1. Kibble-Zurek scaling verified in superfluid helium and liquid crystals.

19.2 LHC Predictions

Prediction 2. The Higgs boson exists and will be discovered at the LHC.
Prediction 3. At TeV scale, partial electroweak symmetry restoration becomes visible.
Prediction 4. If R⁻¹ ~ TeV, Kaluza-Klein excitations appear as heavy particle towers.

19.3 Gravitational Waves

Prediction 5. Stochastic gravitational wave background from cosmological symmetry redistribution.

19.4 CPT Tests

Prediction 6. CPT exactly conserved. Any apparent violation is projection artifact.
CHAPTER 20

The CPT Connection

Theorem 11 (CPT as Total Symmetry Invariance). CPT conservation reflects the First Law: total symmetry unchanged under combined C, P, T. Individual violations represent asymmetric distribution between sectors.
Proposition 11 (CP Violation). Observed CP violation arises from asymmetric brane-bulk coupling. The CKM matrix phase parameterises this asymmetric projection.
PART VII
Philosophical Implications
CHAPTER 21

The Nature of Physical Law

The conventional view treats physical laws as rules governing how things behave. Our framework suggests a different view: physical laws are symmetry constraints. The 'laws' we discover are not external rules imposed on nature but expressions of the symmetry structure of the universe.

Maxwell's equations are the unique equations consistent with U(1) gauge symmetry and Lorentz invariance. Einstein's equations follow from general coordinate invariance. The 'laws' are theorems derivable from symmetry.

The Unreasonable Effectiveness of Mathematics

Wigner noted 'the unreasonable effectiveness of mathematics in the natural sciences'. Our framework suggests an answer: mathematics is effective because it is the language of symmetry, and the universe is fundamentally symmetric.

CHAPTER 22

Symmetry and Reality

22.1 Plato's Cave Revisited

In Plato's allegory, prisoners see only shadows of objects passing before a fire. Our framework makes this allegory physical. We are the prisoners. The 'shadows' are projections of higher-dimensional symmetric objects onto our 3+1D cave wall. The 'fire' is dimensional projection.

22.2 Ontological Status of Symmetry

Is symmetry real, or merely descriptive? We argue strongly for realism: symmetry is the most fundamental aspect of physical reality. Particles are defined by representations of symmetry groups. Forces arise from gauge symmetries. Conservation laws are symmetry consequences. If we strip away symmetry, nothing remains.

This suggests radical ontology: the universe is made of symmetry. Matter, energy, space, and time are all derived from a more fundamental symmetric substrate.

CHAPTER 23

Conclusions

23.1 Summary

ZEROTH LAW Symmetry equivalence is transitive.
FIRST LAW Total symmetry is conserved.
SECOND LAW Dimensional projection reduces apparent symmetry.
THIRD LAW Big Bang created symmetry with spacetime.
FOURTH LAW Symmetry transfer governed by E/D.
CENTRAL THESIS Symmetry is never truly broken - only hidden.

23.2 Achievements

  • Unification: Gauge breaking, phase transitions, arrow of time, matter-antimatter asymmetry - unified under one framework.
  • Reinterpretation: 'Symmetry breaking' becomes redistribution.
  • Connection: Links symmetry to thermodynamics, information theory, and quantum entanglement.
  • Prediction: Testable predictions for Kibble-Zurek, LHC, gravitational waves, CPT tests.

23.3 Final Words

The laws proposed here may require revision. The specific predictions may prove wrong. But the underlying vision - that symmetry is conserved, that asymmetry is projection, that the universe is fundamentally symmetric - seems too beautiful to be entirely false.

Physics has progressed by finding deeper symmetries. At the deepest level, reality is perfectly symmetric. Our world of broken symmetries is a shadow play - beautiful, but a shadow nonetheless.

"To see past the shadows, to glimpse the symmetric reality behind appearances, is perhaps the highest goal of physics."
REFERENCES
[1] E. Noether, "Invariante Variationsprobleme," Nachr. König. Gesellsch. Wiss. Göttingen, pp. 235-257, 1918.
[2] Y. Nambu, "Quasi-particles and gauge invariance," Phys. Rev. 117, 648, 1960.
[3] J. Goldstone, "Field theories with superconductor solutions," Nuovo Cimento 19, 154, 1961.
[4] P. W. Higgs, "Broken symmetries and masses of gauge bosons," Phys. Rev. Lett. 13, 508, 1964.
[5] S. L. Glashow, "Partial symmetries of weak interactions," Nucl. Phys. 22, 579, 1961.
[6] S. Weinberg, "A model of leptons," Phys. Rev. Lett. 19, 1264, 1967.
[7] T. Kaluza, "Zum Unitätsproblem der Physik," Sitz. Preuss. Akad. Wiss., 966, 1921.
[8] O. Klein, "Quantentheorie und fünfdimensionale Relativitätstheorie," Z. Physik 37, 895, 1926.
[9] M. B. Green, J. H. Schwarz, E. Witten, Superstring Theory, Cambridge, 1987.
[10] L. Randall, R. Sundrum, "Large mass hierarchy from small extra dimension," Phys. Rev. Lett. 83, 3370, 1999.
[11] E. A. Abbott, Flatland: A Romance of Many Dimensions, 1884.
[12] G. 't Hooft, "Dimensional reduction in quantum gravity," gr-qc/9310026, 1993.
[13] L. Susskind, "The world as a hologram," J. Math. Phys. 36, 6377, 1995.
[14] J. Maldacena, "Large N limit of superconformal field theories," Adv. Theor. Math. Phys. 2, 231, 1998.
[15] C. Bäuerle et al., "Laboratory simulation of cosmic string formation," Nature 382, 332, 1996.
[16] I. Chuang et al., "Cosmology in the laboratory," Science 251, 1336, 1991.
[17] T. W. B. Kibble, "Topology of cosmic domains and strings," J. Phys. A 9, 1387, 1976.
[18] W. H. Zurek, "Cosmological experiments in superfluid helium?," Nature 317, 505, 1985.
[19] CERN, "LHC Design Report," CERN-2004-003, 2004.
[20] E. P. Wigner, "Unreasonable effectiveness of mathematics," Comm. Pure Appl. Math. 13, 1, 1960.
END OF TREATISE
Subhadip Mitra
[email protected]
Kshitij 2006, IIT Kharagpur