Scaled Affine Quantization of Ultralocal $\varphi^{4}_{2}$ a comparative Path Integral Monte Carlo study with scaled Canonical Quantization

Ultralocal euclidean scalar field quantization, henceforth denoted $\varphi^{r}_{n}$, where $r$ is the power of the interaction term and $n=s+1$ where $s$ is the spatial dimension and $1$ adds imaginary time, such that $r<2$ can be treated by canonical quantization (CQ), while models such that $r>2$ and any $n\geq 2$ are trivial [1, 2, 3, 4, 5]. However, there exists a different approach called affine quantization (AQ) [1, 6, 7] that promotes a different set of classical variables to become the basic quantum operators and it offers different results, such as models for which $r>2$. In particular one can show that while the Fubini-Study metric for the canonical coherent states that evaluates the distance-squared between two infinitesimally close ray-vectors (minimized over any simple phase) leads to a flat space that already involves Cartesian coordinates, in the affine case the Fubini-Study metric describes a Poincaré half plane [8, 7], has a constant negative curvature [9], and is geodesically complete. Unlike a flat plane, or a constant positive curvature surface (which holds the metric of three-dimensional spin coherent states), a space of constant negative curvature can not be visualized in a 3-dimensional flat space. At every point in this space the negative curvature appears like a saddle having an “up curve” in the direction of the rider’s chest and a “down curve” in the direction of the rider’s legs. ¹¹1Of course other types of more complex quantizations may still be possible who involve non-constant curvature surfaces [10, 11].

In the present work we show, with the aid of a Path Integral Monte Carlo (PIMC) analysis, that $r=4$ and $n=2$ can be acceptably quantized using scaled affine quantization which had been previously successfully used for the covariant case [12, 13, 14, 15, 16, 17, 18, 19]

Being the current study in a lower, therefore unphysical, spacial dimension it nonetheless allowed us to get closer to the continuum limit than it was feasible for the physically relevant four-dimensional case on the computer due to the rapid increase of necessary lattice points as dimensionality is increased. Therefore this work can indirectly give us a better understanding of the physically relevant case which had been already preliminarily studied by us in its covariant version [13]. Interestingly, the triviality of the scaling limits of the canonical Ising and covariant self-interacting scalar field models in four dimensions has been rigorously demonstrated recently [20].

The quantum affine operators are the scalar field $\hat{\varphi}(x)=\varphi(x)$ and the dilation operator ³³3Since $\varphi(x)\neq 0$, that means $\pi^{\dagger}\neq\pi$ so, to make that clear we should say that $\hat{\kappa}(x)\equiv[\hat{\varphi}(x)\hat{\pi}(x)+\hat{\pi}^{\dagger}(x)\hat{\varphi}(x)]/2$ to make sure that $\hat{k}^{\dagger}=\hat{k}$. But $\hat{\pi}^{\dagger}\hat{\varphi}=\hat{\pi}\hat{\varphi}$ because in that case $\hat{\pi}^{\dagger}$ acts like $\hat{\pi}$ thanks to having $\hat{\pi}$ acting on $\hat{\varphi}$. $\hat{\kappa}(x)=[\hat{\varphi}(x)\hat{\pi}(x)+\hat{\pi}(x)\hat{\varphi}(x)]/2$ where the momentum operator is $\hat{\pi}(x)=-i\hbar\delta/\delta\varphi(x)$. Accordingly for the self adjoint kinetic term $\hat{\kappa}(x)\hat{\varphi}(x)^{-2}\hat{\kappa}(x)=\hat{\pi}(x)^{2}+(3/4)\hbar\delta(0)^{2s}\varphi(x)^{-2}$ and one finds for the quantum Hamiltonian operator

As in previous works [16, 17, 19] we use the scaling $\pi\rightarrow a^{-s/2}\pi,\varphi\rightarrow a^{-s/2}\varphi,g\rightarrow a^{s(r-2)/2}g$, which is necessary ⁴⁴4 Note that from a physical point of view one never has to worry about the mathematical divergence since the lattice spacing will necessarily have a lower bound. For example at an atomic level one will have $a\gtrsim 1$Å. In other words the continuum limit will never be a mathematical one. to eliminate the Dirac delta factor $\delta(0)=a^{-1}$ divergent in the continuum limit $a\to 0$. Of course for $r>2$ the rescaled coupling constant, $g$, vanishes in the continuum limit since $a\to 0$, therefore we expect no difference between the interacting and the free model in such a limit. The theory considers a real scalar field $\varphi$ taking the value $\varphi(x)$ on each site of a periodic, hypercubic, $n$-dimensional lattice of lattice spacing $a$, our ultraviolet cutoff, and periodicity $L=Na$. The affine action for the field, ${\cal S}^{\prime}=\int\bar{H^{\prime}}\,dx_{0}$ (with $x_{0}=ct$ where $c$ is the speed of light constant and $t$ is imaginary time), with $\bar{H^{\prime}}$ the semi-classical Hamiltonian corresponding to the one of Eq. (3), is then approximated by

where $e_{\mu}$ is a vector of length $a$ in the $+\mu$ direction. Respect to the previously considered covariant case [12, 13, 14, 15, 16, 17, 18, 19], being now absent derivatives with respect to space, the field is allowed to be discontinuous in space (but will still be continuous in time).

The corresponding canonical action, ${\cal S}=\int\bar{H}\,dx_{0}$, is then approximated by

In this work we are interested in reaching the continuum limit by taking $Na$ fixed and letting $N\to\infty$ at fixed volume $L^{s}$ and absolute temperature $T=1/k_{B}L$ with $k_{B}$ the Boltzmann’s constant.

The vacuum expectation value of an observable ${\cal O}[\varphi]$ will then be given by the following expression

where the functional integrals will be calculated on a lattice using the path integral Monte Carlo method as explained further on.

We performed PIMC [21, 22, 23, 24] for the action of Eq. (4) with $r=4$ and $n=2$. In particular we calculated the renormalized coupling constant $g_{R}$ and mass $m_{R}$ defined in Eqs. (11) and (13) of [13] respectively, measuring them in the path integral MC through vacuum expectation values like in Eq. (6). In particular $m_{R}^{2}=p_{0}^{2}\langle|\tilde{\varphi}(p_{0})|^{2}\rangle/[\langle\tilde{\varphi}(0)^{2}\rangle-\langle|\tilde{\varphi}(p_{0})|^{2}\rangle]$ and at zero momentum $g_{R}=[3\langle\tilde{\varphi}(0)^{2}\rangle^{2}-\langle\tilde{\varphi}(0)^{4}\rangle]/\langle\tilde{\varphi}(0)^{2}\rangle^{2}$, where $\tilde{\varphi}(p)=\int d^{n}x\;e^{ip\cdot x}\varphi(x)$ is the Fourier transform of the field and we choose the 2-momentum $p_{0}$ with the zero component equal to $2\pi/Na$ and the other component equal to zero. Since the integration variables in Eq. (6) are $N^{n}$, being able to choose $n=2$ allowed us to greatly speed up the calculations compared to our previous covariant studies for $n>2$ [12, 13, 14, 15, 16, 17, 18, 19] and this made possible to push ourselves closer to the continuum limit, to bigger $N$.

Following Freedman et al. [2], for each $N$ and $g$, we adjusted the bare mass $m$ in such a way to maintain the renormalized mass approximately constant, $m_{R}\approx 3$, to within a few percent (in all cases less than $20\%$), and we measured the renormalized coupling constant $g_{R}$ for various values of the bare coupling constant $g$ at a given small value of the lattice spacing $a=1/N$ (this corresponds to choosing an absolute temperature $k_{B}T=1$ and a fixed length $L=1$). Note that in the CQ case it was necessary to choose imaginary bare masses for $g>0$. With $Na$ and $m_{R}$ fixed, as $a$ was made smaller, whatever change we found in $g_{R}m_{R}^{n}$ as a function of $g$ could only be due to the change in $a$. We generally found that a depression in $m_{R}$ produced an elevation in the corresponding value of $g_{R}$ and viceversa; for this reason it is convenient to define an alternative renormalized coupling constant less sensitive to small variations of $m_{R}$, namely $g_{R}(m_{R})^{n}$ (see [2]). The results are shown in Fig. 1 for the scaled canonical action (5) and the scaled affine action (4) in natural units $c=\hbar=k_{B}=1$, where, following Freedman et al. [2] we decided to compress the range of $g$ for display, by choosing the horizontal axis to be $g/(50+g)$. The constraint $m_{R}\approx 3$ was not easy to implement since for each $N$ and $g$ we had to run the simulation several times with different values of the bare mass $m$ in order to determine the value which would satisfy the constraint $m_{R}\approx 3$. This was the main source of uncontrolled uncertainty in the data.

In our simulations we used $10^{8}$ MC steps where in each step we attempt to move once all the $N^{n}$ fields variables of integration through the Metropolis algorithm [21]. We used block averages and estimated the statistical errors using the jakknife method (described in Section 3.6 of [25]) to take into account of the correlation time of the simulations. We always adjusted the field displacement in the random walk so to keep the acceptance ratios as close as possible to $1/2$.

Comparing the results for the scaled canonical and affine action we can see how the renormalized coupling constant of the two approaches behaves very similarly at $g\neq 0$, ⁵⁵5Comparing with the previous covariant studies [12, 13, 14, 15, 16, 17, 18, 19] we can now say that removing the gradient term leads to a wilder behavior of the paths which could complicate finding any difference between CQ and AQ. but in a neighborhood of $g=0$ the affine version remains far from zero in the continuum limit when the ultraviolet cutoff is removed ($Na=1$ and $N\to\infty$). The decrease of the renormalized coupling $g_{R}$ for increasing $N$ has to be expected, both for the CQ and the AQ cases, due to the use we made of the scaling $g\rightarrow a^{s(r-2)/2}g$ which makes the model a “free” one in the continuum limit, $a\to 0$, when $r>2$. Of course the scaling we used has just a mathematical and not a physical justification (see footnote 4). In particular the scaling permits to have non diverging field expectation values in the AQ approach. Therefore as already observed in several previous covariant studies [12, 13, 14, 15, 18] we expect that also in this ultralocal case the AQ approach gives rise to a “non-free” field theory contrary to what happens for the CQ approach for $r>2$. This success of affine quantization to produce a well-defined, renormalizable, nontrivial, “non-free” quantum field theory is one of its merits and benefits.

During our simulations we kept under control also the vacuum expectation value of the field which in all cases was found to vanish in agreement with the fact that the symmetry $\varphi\to-\varphi$ of the scaled canonical action is preserved in the scaled affine case. The random walk in the field is always able to tunnel through the barrier at $\varphi=0$ due to the affine effective term, $\frac{3}{8}(\hbar/\varphi)^{2}$, in the interaction. This is a consequence of working at finite $N$ and we expect the symmetry to be spontaneously broken in the continuum $N\to\infty$ limit ⁶⁶6Once again this is only possible in a mathematical world but not in the physical (see footnote 4). when the point at $\varphi=0$ is excluded. Our results also show how the sum rules $g_{R}\to 0$ and $m_{R}\to m$ for $g\to 0$ are satisfied for CQ as it should for any gaussian weighting factor $\exp(-{\cal S}[\varphi])$ for any $N$.

In conclusion we performed a path integral Monte Carlo study of the properties (mass and coupling constant) of the renormalized ultralocal euclidean scalar field theory $\varphi^{4}_{2}$ quantized through scaled affine and canonical quantization. Our results confirm the theoretical expectation for a “free” theory in the continuum limit. This is merely a consequence of the chosen scaling. As in previous works on covariant theories we expect that also in this ultralocal case the un-scaled AQ approach gives rise to a “non-free” field theory contrary to what happens for the un-scaled CQ approach for $r>2$. Indeed already for the scaled version in the AQ theory the renormalized coupling does not seem to go towards zero at least when the bare coupling is zero when one approaches the continuum limit, as stems from our path integral Monte Carlo results. This means that a “free” scaled AQ theory is profoundly different from a “free” scaled CQ one: the former is therefore non-trivial and renormalizable and the latter is trivial and non-renormalizable.