Conjuncted: All Sciences Are Still Externalist.

Constructor Theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations (Deutsch). Accordingly, a task A is said to be possible (A✓) if the laws of nature impose no restrictions on how (accurately) A could be performed, nor on how well the agents that are capable of approximately performing it could retain their ability to do so. Otherwise A is considered to be impossible (A✘). Deutsch argues that in both quantum theory and general relativity, “time is treated anomalously”. He sees the problem in that time is not among the entities to which both theories attribute “objective existence (namely quantum observables and geometrical objects respectively), yet those entities change with time”. Is not this an interesting appraisal? According to him “there is widespread agreement that there must be a way of treating time ‘intrinsically’ (i.e. as emerging from the relationships between physical objects such as clocks) rather than ‘extrinsically’ (as an unphysical parameter on which physical quantities somehow depend).” However, Deutsch reckons, it would be difficult to accommodate this in the prevailing conception, “every part of which (initial state; laws of motion; time-evolution) assumes that extrinsic status”. According to Constructor Theory it is “both natural and unavoidable to treat both time and space intrinsically: they do not appear in the foundations of the theory, but are emergent properties of classes of tasks …”. Note that the differentiation between “intrinsic” and “extrinsic” has different meaning than the one used in phenomenological philosophy.

Other projections, such as the Circular Theory (Yardley) and the Structural Theory of Everything (Josephson) also question the laws underlying the foundations of physics. However, as long as they consider placing events “over time” extrinsically, they are all of the same kind: non-phenomenological. (Note that duration has two definitions: endo and exo). This is a very interesting conclusion in Steven M. Rosen’s words:

The point may be that, in physics, time is treated ‘extrinsically’ because it does not lend itself to being formulated in objectivist terms, i.e., as something that is limited to the context of the objectified physical world.

Therefore, the terms “intrinsic” and “extrinsic” in Deutsch’s objectivist usage are both “extrinsic” in the phenomenological sense: both are limited to a physicalistic paradigm that excludes the internal perspective of a lived (bodily) subjectivity or an agent-dependent reality (Rössler), a continuing durée (Bergson). This appears to be the key to all the “trouble with physics” (Smolin), and not only with physics.  In their 2012 paper No entailing laws, but enablement in the evolution of the biosphere Longo, Montévil and Kauffman claim that biological evolution “marks the end of a physics world view of law entailed dynamics” (Longo et al.). They argue that the evolutionary phase space or space of possibilities constituted of interactions between organisms, biological niches and ecosystems is “ever changing, intrinsically indeterminate and even (mathematically) unprestatable”. Hence, the authors’ claim that it is impossible to know “ahead of time the ‘niches’ which constitute the boundary conditions on selection” in order to formulate laws of motion for evolution. They call this effect “radical emergence”, from life to life. In their study of biological evolution, Longo and colleagues carried close comparisons with physics. They investigated the mathematical constructions of phase spaces and the role of symmetries as invariant preserving transformations, and introduced the notion of “enablement” to restrict causal analyses to Batesonian differential cases. The authors have shown that mutations or other “causal differences” at the core of evolution enable the establishment of non-conservation principles, in contrast to physical dynamics, which is largely based on conservation principles as symmetries. Their new notion of “extended criticality” also helps to understand the distinctiveness of the living state of matter when compared to the non-animal one. However, their approach to both physics and biology is also non-phenomenological. The possibility for endo states that can trigger the “(genetic) switches of mutation” has not been examined in their model. All sciences are (still) externalist.

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Conjuncted: Canonical Hamiltonian as Road to Gauge Freedom.

A basic ingredient of the canonical formalism is the Hamiltonian function. In the case of a regular theory (that is, with a non-singular Hessian matrix) it defines, by use of the Poisson bracket, the vector field that generates the time evolution – the dynamics – in phase space. The Hamiltonian is given in that case as the projection to phase space of the Lagrangian energy E = ∂L/∂q ̇ − L.

This procedure to define the Hamiltonian will still work in the singular case if the energy satisfies the conditions of projectability (7). Indeed we can readily check that Γ_μE = 0, so we have a canonical Hamiltonian H_c, defined as a function on phase space whose pullback is the Lagrangian energy, FL ∗ H_c = E. It was Dirac that first realized in the general setting of constrained systems that a Hamiltonian always existed.

There is a slight difference, though, from the regular case, for now there is an ambiguity in the definition of Hc. In fact, since FL ∗ φ_μ = 0, many candidates for canonical Hamiltonians are available, once we are given one. In fact, H_c + v^μφ_μ — with v^μ(q, q ̇; t) arbitrary functions and with summation convention for μ – is as good as H_c as a canonical Hamiltonian. This “slight difference” is bound to have profound consequences: it is the door to gauge freedom.

Time-Evolution in Quantum Mechanics is a “Flow” in the (Abstract) Space of Automorphisms of the Algebra of Observables

In quantum mechanics, time is not a geometrical flow. Time-evolution is characterized as a transformation that preserves the algebraic relations between physical observables. If at a time t = 0 an observable – say the angular momentum L(0) – is defined as a certain combination (product and sum) of some other observables – for instance positions X(0), Y (0) and momenta PX (0), PY (0), that is to say

L(0) = X (0)P_Y (0) − Y (0)P_X (0) —– (1)

then one asks that the same relation be satisfied at any other instant t (preceding or following t = 0),

L(t) = X (t)P_Y (t) − Y (t)P_X (t) —– (2)

The quantum time-evolution is thus a map from an observable at time 0 to an observable at time t that preserves the algebraic form of the relation between observables. Technically speaking, one talks of an automorphism of the algebra of observables.

At first sight, this time-evolution has nothing to do with a flow. However there is still “something flowing”, although in an abstract mathematical space. Indeed, to any value of t (here time is an absolute parameter, as in Newton mechanics) is associated an automorphism α_t that allows to deduce the observables at time t from the knowledge of the observables at time 0. Mathematically, one writes

L(t) = α_t(L(0)), X(t) = α_t(X(0)) —– (3)

and so on for the other observables. The term “group” is important for it precisely explains why it still makes sense to talk about a flow. Group refers to the property of additivity of the evolution: going from t to t′ is equivalent to going from t to t₁, then from t₁ to t′. Considering small variations of time (t′−t)/n where n is an integer, in the limit of large n one finds that going from t to t′ consists in flowing through n small variations, exactly as the geometric flow consists in going from a point x to a point y through a great number of infinitesimal variations (x−y)/n. That is why the time-evolution in quantum mechanics can be seen as a “flow” in the (abstract) space of automorphisms of the algebra of observables. To summarize, in quantum mechanics time is still “something that flows”, although in a less intuitive manner than in relativity. The idea of “flow of time” makes sense, as a flow in an abstract space rather than a geometrical flow.

What Drives Investment? Or How Responsible is Kelly’s Optimum Investment Fraction?

A reasonable way to describe assets price variations (on a given time-scale) is to assume them to be multiplicative random walks with log-normal step. This comes from the assumption that growth rates of prices are more significant than their absolute variations. So, we describe the price of a financial assets as a time-dependent multiplicative random process. We introduce a set of N Gaussian random variables x_i(t) depending on a time parameter t. By this set, we define N independent multiplicative Gaussian random walks, whose assigned discrete time evolution is given by

p_i(t+1) = e^x_i(t)p_i(t) —– (1)

for i = 1,…,N, where each x_i(t) is not correlated in time. To optimize an investment, one can choose different risk-return strategies. Here, by optimization we will mean the maximization of the typical capital growth rate of a portfolio. A capital W(t), invested into different financial assets who behave as multiplicative random walks, grows almost certainly at an exponential rate ⟨ln W (t+1)/W (t)⟩, where one must average over the distribution of the single multiplicative step. We assume that an investment is diversified according to the Kelly’s optimum investment fraction, in order to maximize the typical capital growth rate over N assets with identical average return α = ⟨e^x_i(t)⟩ − 1 and squared volatility ∆ = ⟨e^2x_i(t)⟩ − ⟨e^x_i(t)⟩². It should be noted that Kelly capital growth criterion, which maximizes the expected log of final wealth, provides the strategy that maximizes long run wealth growth asymptotically for repeated investments over time. However, one drawback is found in its very risky behavior due to the log’s essentially zero risk aversion; consequently it tends to suggest large concentrated investments or bets that can lead to high volatility in the short-term. Many investors, hedge funds, and sports bettors use the criterion and its seminal application is to a long sequence of favorable investment situations. On each asset, the investor will allocate a fraction f_i of his capital, according to the return expected from that asset. The time evolution of the total capital is ruled by the following multiplicative process

W(t+1) = [1 + ∑_i=1^Nf_i(e^x_i(t) -1)] W(t) —– (2)

First, we consider the case of an unlimited investment, i.e. we put no restriction tothe value of ∑_i=1^Nf_i. The typical growth rate

V_typ = ⟨ln[1+  ∑_i=1^Nf_i(e^x_i -1)]⟩ —– (3)

of the investor’s capital can be calculated through the following 2^nd-order expansion in e^x_i -1, if we assume that fluctuations of prices are small and uncorrelated, that seems to be quite reasonable

V_typ ≅ ∑_i=1^Nf_i(⟨e^x_i⟩ – 1) – f_i²/2(⟨e^2x_i⟩ – 2⟨e^x_i⟩ + 1 —– (4)

By solving d/df(V_typ = 0), it easy to show that the optimal value for f_i is f_i^opt (α, Δ) = α / (α² + Δ) ∀ i. We assume that the investor has little ignorance about the real value of α, that we represent by a Gaussian fluctuation around the real value of α. In the investor’s mind, each asset is different, because of this fluctuation α_i = α + ε_i. The ε_i are drawn from the same distribution, with ⟨ε_i⟩ = 0 as errors are normally distributed around the real value. We suppose that the investor makes an effort E to investigate and get information about the statistical parameters of the N assets upon which he will spread his capital. So, his ignorance (i.e. the width of the distribution of the ε_i) about the real value of α_i will be a decreasing function of the effort “per asset” E ; more, we suppose that an even infinite effort will not make N this ignorance vanish. In order to plug these assumptions in the model, we write the width of the distribution of ε as

⟨ε²_i⟩ = D₀ + (N/E)^γ —– (5)

with γ > 0. As one can see, the greater is E, the more exact is the perception, and better is the investment. D₀ is the asymptotic ignorance. All the invested fraction f^opt (α_i, Δ) will be different, according to the investor’s perception. Assuming that the ε_i are small, we expand all f_i(α + ε_i) in equation 4 up to the 2^nd order in ε_i, and after averaging over the distribution of ε_i, we obtain the mean value of the typical capital growth rate for an investor who provides a given effort E:

V_typ^– = N[A − (D₀ + (N/E)^γ )B] —– (6)

where

A = (α (3Δ – α²))/(α² + Δ)³ B = -(α² – Δ)²/2(α² + Δ)³ —– (7)

We are now able to find the optimal number of assets to be included in the portfolio (i.e., for which the investment is more advantageous, taken into account the effort provided to get information), by solving d/dNV_typ^– = 0, it is easy to see that the number of optimal assets is given by

N^opt(E) = E {[(A – D₀]/(1 + γ)B}^1/γ —– (8)

that is an increasing function of the effort E. If the investor has no limit in the total capital fraction invested in the portfolio (so that it can be greater than 1, i.e. the investor can invest more money than he has, borrowing it from an external source), the capital can take negative values, if the assets included in the portfolio encounter a simultaneous negative step. So, if the total investment fraction is greater than 1, we should take into account also the cost of refunding loss to the bank, to predict the typical growth rate of the capital.

Phenomenological Model for Stock Portfolios. Note Quote.

The data analysis and modeling of financial markets have been hot research subjects for physicists as well as economists and mathematicians in recent years. The non-Gaussian property of the probability distributions of price changes, in stock markets and foreign exchange markets, has been one of main problems in this field. From the analysis of the high-frequency time series of market indices, a universal property was found in the probability distributions. The central part of the distribution agrees well with Levy stable distribution, while the tail deviate from it and shows another power law asymptotic behavior. In probability theory, a distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. The scaling property on the sampling time interval of data is also well described by the crossover of the two distributions. Several stochastic models of the fluctuation dynamics of stock prices are proposed, which reproduce power law behavior of the probability density. The auto-correlation of financial time series is also an important problem for markets. There is no time correlation of price changes in daily scale, while from more detailed data analysis an exponential decay with a characteristic time τ = 4 minutes was found. The fact that there is no auto-correlation in daily scale is not equal to the independence of the time series in the scale. In fact there is auto-correlation of volatility (absolute value of price change) with a power law tail.

Portfolio is a set of stock issues. The Hamiltonian of the system is introduced and is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. The interaction coefficients between two stocks are phenomenologically determined by empirical data. They are derived from the covariance of sequences of up and down spins using fluctuation-response theorem. We start with the Hamiltonian expression of our system that contain N stock issues. It is a function of the configuration S consisting of N coded price changes S_i (i = 1, 2, …, N ) at equal trading time. The interaction coefficients are also dynamical variables, because the interactions between stocks are thought to change from time to time. We divide a coefficient into two parts, the constant part J_ij, which will be phenomenologically determined later, and the dynamical part δJ_ij. The Hamiltonian including the interaction with external fields h_i (i = 1,2,…,N) is defined as

H [S, δ, J, h] = ∑_<i,j>[δJ_ij²/2Δ_ij – (J_ij + δJ_ij)S_iS_j] – ∑_ih_iS_i —– (1)

The summation is taken over all pairs of stock issues. This form of Hamiltonian is that of annealed spin glass. The fluctuations δJ_ij are assumed to distribute according to Gaussian function. The main part of statistical physics is the evaluation of partition function that is given by the following functional in this case

Z[h] = ∑_{si} ∫∏_<i,j> dδJ_ij/√(2πΔ_ij) e^{-H [S, δ, J, h]} —– (2)

The integration over the variables δJij is easily performed and gives

Z[h] = A ∑_{si} e^{-H_eff[S, h]} —– (3)

Here the effective Hamiltonian H_eff[S,h] is defined as

H_eff[S, h] = – ∑_<i,j>J_ijS_iS_j – ∑_ih_iS_i —– (4)

and A = e(1/2 ∆_ij) is just a normalization factor which is irrelevant to the following step. This form of Hamiltonian with constant J_ij is that of quenched spin glass.

The constant interaction coefficients J_ij are still undetermined. We use fluctuation-response theorem which relates the susceptibility χ_ij with the covariance C_ij between dynamical variables in order to determine those constants, which is given by the equation,

χ_ij = ∂m_i/∂h_j |_h=0 = C_ij —– (5)

Thouless-Anderson-Palmer (TAP) equation for quenched spin glass is

m_i =tanh(∑_jJ_ijm_j + h_i – ∑_jJ_ij²(1 – m_j²)m_i —– (6)

Equation (5) and the linear approximation of the equation (6) yield the equation

∑_k(δ_ik − J_ik)C_kj = δ_ij —– (7)

Interpreting C_ij as the time average of empirical data over a observation time rather than ensemble average, the constant interaction coefficients J_ij is phenomenologically determined by the equation (7).

The energy spectra of the system, simply the portfolio energy, is defined as the eigenvalues of the Hamiltonian H_eff[S,0]. The probability density of the portfolio energy can be obtained in two ways. We can calculate the probability density from data by the equation

p(E) ΔE = p(E – ΔE/2 ≤ E ≤ E + ΔE/2) —– (8)

This is a fully consistent phenomenological model for stock portfolios, which is expressed by the effective Hamiltonian (4). This model will be also applicable to other financial markets that show collective time evolutions, e.g., foreign exchange market, options markets, inter-market interactions.